How good are you at diy plumbing work?

My suggestion would be to cut the pipe at the the point where than connection is that shrinks the size of the pipe. Then I would sweat on the male counterpart to that hose you have. I would enlarge the hole in the cabinet and feed the hose thru there.

Don't use a hacksaw, you won't get a clean/straight cut. You need a tubing cutter. You'll need to clean the end of the pipe with some steel wool so that it is shiny. You'll need some solder flux, solder, and a torch.

A plumber could do this fairly quickly if you don't want to do it yourself.

There is a very simple DIY soluton that willl not require any soldering.
First of all, as was mentioned above, DO NOT cut copper pipe or tubing with a hacksaw. In nearly all cases the stress of cutting copper with a hacksaw results in getting the pipe ever so slightly out of round, however it is enough that it is sometimes nearly impossible to get a fitting on the copper again, and even if your lucky enough to get the fitting on there is an extremely high risk that the joint will not properly seal. Copper should always be cut with a tubing cutter. When working in a tight spot such as you have there a "mini tubing cutter" is the ideal tool. On bit of advice here. You will find small red mini tubing cutters in all hardware and home supply stores for about $5. In my humble opion those are at best junk. For just a bit more ($10 to $12 range) you can get a mini tubing cutter made by Imperial Tool Company, Superior Tool Company or Rigid Tool company and you will have a pro quality tool that will hold up for years under DIY conditions.
Begin by turning the water off with the ball valve, then using a 2" wide strip of emery cloth wrapped around the pipe and pull it back and forth like a shoe shine rag to clean the tarnish off the copper.
Using a tubing cutter cut the copper pipe at the point I indexed on the attached illustration.
Now this might seem a bit confusing. The 1/2" copper extending out of the ball valve at the cut point is 1/2" copper pipe, which means it has a 1/2" inside diameter (ID). The outside diameter of 1/2" copper pipe is 5/8" and copper tubing is rated by its outside diameter (OD) so a 5/8" tubing fitting will fit directly on 1/2" copper pipe.
You will need a compression type reducing union. Most reducing unions are rated for copper tubing therefore the one you need would be a 3/8" x 5/8 reducing union. Some manufactures now rate them as 3/8" by 5/8" tubing & 1/2" copper pipe.
To install you will remove the compression nut on the large end of the union. You will find a brass compression ferrule under the nut. (looks like a cheap mans wedding ring). Slide the nut on the copper with the open side of the nut facing the cut end of the pipe, then slide the compression ferrule on the pipe. Now slide the union on the pipe until you feel the copper pipe hit the internal stop insuring it is on all the way, then slide the compression ferrule towards the union and tighten the nut on the union. You should hold the union steady with a pair of channel locks while tightening the nut with an adjustable wrench. Make sure the nut is tight.
Now remove the compression nut and ferrule from the small end of the union and you will have a 3/8" male thread extending out. The line from your dishwasher will screw directly on that thread.
Do Not put teflon tape or thread joint compound one any of the threads. They are all mechanical joints that are intended to be dry fit.
Another hint, If you cannot find a reducing union check in the area where they have the angle stop and line stop valves for fixtures. You could use a compression type straight through line stop in the same manner as the compression union. The only difference is that you would then have a redundant valve but if left open it is no problem and probably a compression straight valve would be slightly cheaper.

this is kinda what Pup is talking about. You would need to get one the right size for you.

Here is a link that might be useful: Union

Thanks for being thorough lazypup! Love the diagram!

After some careful measuring, we realized that the union provided for us is for a 3/8" to 1" since the dishwasher hose is 1" , like a garden hose. We discovered that the current coper tubing (that you labeled 3/8) doesn't fit into the tubing reducer, though it could have been flanged out because of the previous compression joint from the old dishwasher.

Nonetheless, you've helped us figure out what we need to do. We're going to need a 5/8" to 1" "reducer" and then we'll thread the dishwasher hose into the cabinet. Hopefully that will take care of it, and save us a few labor hours from a plumber.

The line from the dishwasher IS NOT a 1" water line. The large protrusion on the end of the line is an electrically operated solenoid valve and the 1" diameter plastic tubing to the dishwasher is a flexible conduit which encloses the water line and the electrical control wires for the solenoid.

The vinyl water line inside that conduit is at best 3/8" ID, which is the industry standard for a dishwasher supply line.