Yes, remove the stalk so the plant can put it's energy into a new stalk and leaves.

Here is a picture of a rebloomed spike on a hybrid Phalaenopsis. Not a thing of beauty. However, different suggestions apply to species and some primary hybrids. This subject is discussed in some detail in one of the FAQ's (frequently asked questions)
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So, do I cut it back to the plant itself or somewhere higher up on the flower stalk?

If you don't want it to rebloom on the same stalk, than cut it of just above the first leaf nod and dust the cut with cinnamon to disinfect it. Wait to do this until all the flowers have fallen of. Good luck!

Cut it down as far as you can reach, no sense in leaving anything behind. And no need to do cinnamon either. The spike will die faster than any infection can take hold.

Aside: I'm not personally a fan of cinnamon, it has anti-rooting properties and I cringe every time I hear people suggest it, especially suggesting a dusting on the roots *D'OH!* Plenty of people use it, but I don't and I haven't had any infections as a result of cuts or damage in the decade I've been growing.

Amen, terpguy!

Jane

Alright, guys. Thanks for the help!

-Sam

I could not agree more with the cinnamon thing! People go crazy witht he stuff!