Yes, but...you can. I see it in San Antonio all the time. The ONLY secret is to keep the St Augustine watered once a week (very deeply if possible). The secondary secret to allow full coverage for St Aug is to mow at your mower's highest settinng. In those circumstances, in San Antonio, St Aug wins out every time. But if you stop watering for a couple weeks then suddenly the entire area will be bermuda. I have done this myself and gone back and forth. My driveway in the back is stone. The St Aug escaped into it years ago and the bermuda volunteered. We never water the area except for my experiments with shifting back and forth from one grass to the other. In my absence and a drought year, the St Aug is all gone now. Oh and I always forget about the load of dirt the city used to backfill a repair on my water meter. It grew nutgrass and bermuda, but the St Aug filled back in. I did nothing and now it is all St Aug. Here's the problem in a nutshell. When there is a drought, if you don't water bermuda, it goes dormant and will return at the first sign of moisture. If you don't water St Augustine, it goes dead and will not return, period. So if you have an unirrigated and mixed lawn, the bermuda will win. But at the same time, bermuda cannot tolerate any shade. That's why you have areas of St Aug now. They are in the shade of something (fence, building, tree, etc.). Since bermuda is so sun loving, anything you can do to shade it will give the edge to the St Augustine. Mowing the grass at the highest setting will provide enough shade from the St Augustine to choke out the bermuda. St Augustine will spread 10 feet in all directions in one season. A few strategically placed flats (about $1 each at most SA nurseries) should give you a full St Aug lawn by October. If you are going to get on an organic program, now is a good time to start with fertilizer. If you are on a chemical program, do not fertilize until after you mow real grass (not weeds) for the second time. Waiting like that ensures you are putting fertilizer on active grass roots and not just weed roots. My lawn could not be easier to care for. I changed to organic in 2002 and life got real easy. All I have to do is water deeply once a week, mow high (as high as my wife will let me), and fertilize on the federal holidays. I only get weeds in the back where the dog and kids live. There is a FAQ on organic lawn care over on the Organic Gardening forum. It describes an inexpensive, clean, and very effective program.

A picture would be worth a thousand words here. Bringing in soil might be good if you suspect the original soil has been washed away for some reason. Some trees will react to hard soil by sending their roots out at the surface. Then as the tree ages, the roots become larger and expose themselves above the surface. If that is what has happened, then adding soil on top of the natural grade will cause you drainage problems and could cost you extensively if water pours into your basement (worst case). Adding soil should be done with considerable forethought as to the consequences. Another thought on adding soil is to keep it lower than the root bark on the trees. Tree roots flare out from the base of the trunk. If you bury the root flare completely, then you are burying tree bark. Tree bark will often rot when it is below the soil level. It is best if you leave some root flare above the soil level to be sure you are not burying tree bark. Tilling is a bad idea for many reasons. One of the obvious ones is that you have roots everywhere. You can't till tree roots. A much less obvious reason is that tilling destroys soil structure by fluffing it up. Once it is fluffy it has to settle again. That process takes 3 years. When it finally settles it will settle unevenly leaving your surface bumpy. Grass seed should never be mixed in with surface soil. It always goes on top of the surface. Many people recommend "scratching" it into the surface but that is not necessary. Nor is covering it with straw. All you need to do is loosen the soil at the surface (1/16 inch deep) before you apply the seed, then apply the seed, and finally roll the seed down with a water filled roller (rented). Seed should remain at the surface. If you insist on straw, as goren said, be careful not to use hay. Hay is forage for animals and has seed in it. Straw is the left over after the seed is threshed out. New seed must be watered 3x per day at breakfast, lunch, and dinner for about 10 minutes. The idea is to provide enough consistent moisture for the seed to swell and open (germinate). Unfortunately these are the exact conditions that crabgrass seed needs. For this reason grass seeding is most successfully done in the fall after crabgrass has stopped germinating. Also grass sprouted now will have weak roots that cannot stand up to the summer heat and dry weather. That might not be that much of an issue in your area. Don't assume you have clay. Fill a mayonnaise jar half full of dirt. Then fill the jar completely with water and a drop of soap. Swish it around and let it settle. You will see rocks and sand settling immediately to the bottom of the jar. In an hour all the silt will have settled. At that time if you have clay, the water will be cloudy. Let it sit another day and see if the water clears up. If it clears then you do not have clay. Clay will cloud the water for days and days. There are several conditions of minerals that cause soil to seem to be clay when in fact there is no clay at all. Don't be too surprised by this. Adding gypsum without doing this test and a soil chemistry test can be a mistake. I disagree with goren on mowing height. The only grasses that need to be mowed lower than 3 inches are bermuda, centipede, bentgrass, and some zoysias. Kentucky bluegrass (which you can't grow in shade) should be no lower than 3 inches. Fescues and all the other grass types should be mowed back to 4 inches. Usually this is a mower's highest setting. Grass grown in the shade needs to be longer simply so it can photosynthesize. Can you post pictures showing the tree roots and one from a distance showing the trees with the canopy?

You need to study the BB then ask questions. There is a lot of information available here and other places. Your lawn may take off pretty soon. If you are not ready to mow it correctly you will hate the way it looks. Bermuda is considered a weed by anyone who has another type of grass. It takes over lawns and is tough to kill. But as a primary lawn it is awesome. Your watering plan to help establish your sod is going to make your lawn grow. You may need to mow upwards of 3-4 times per week (that's not a typo). I use growth regulators to curb upward growth of my lawn and still have to mow every 3 days, but I have a more aggressive type of bermuda. Get on a good mowing, pgr (plant growth regulator), pre-emergent, feeding schedule and that lawn will get a very lush, dark green color, and get very dense.

The type of bermuda seed is on the back of the bag if you still have it. Completely agree with RJ on mechanical stimulation of bermuda. I'm currently plotting my strategy for next year. A number of golf courses I researched verticut and sand every 2 weeks. The core aerate every month. With that in mind I'm looking to verticut at least three times and aerate x2 next year. From my research, if the lawn is healthy, stable and mature (2-3 years old) you can verticut of aerate anytime during the growing season. So we could in theory be safe upwards of a week or two before out first frost dates. This Winter is my lawn's 1st so I don't know if/when my lawn will go dormant. Your backyard look good. You should do a blanket application of Celsius and Prodiamine (or Dimension) very soon. Kill those weeds and let the bermuda spread a little more be it gets cold. Make sure to do pre-emergent on the front lawn as well. You have a good mower. Check your blade's sharpness and the rolliness of you roller (does it freely/easily spin). Your grass may be laying down longways and not being cut. Even though the mower is set at 1. 25" the actual length of the grass may be long. If you got a few hundred dollars laying around getone of these, otherwise any dime-store ruler will do. Make sure you lightly comb the grass and stand it up before you measure it. You probably can't cut in 3-4 different direction up by that fence which may lead to the grass laying down lengthwise.