Can I replace the aluminum cable of a single circuit with romex?
I want to use cable with copper conductors instead of aluminum if at all possible.

You really can't buy aluminum branch cable anymore, so copper is your only route here.
I would suggest going the extra mile here and buying some Alumiconns and converting to copper (they can be bought in 2-packs if you don't want to buy more than you need right now). Contrary to batplatsterson, I've actually found them decently easy to work with AND you permanently fix the problem so you can use copper wiring and fixtures anytime afterward. What he's likely thinking of are the Noalox-filled wiring nuts (they're purple), and those don't prevent the arcing risk of twisting the wires together.
That having been said, I did have one outdoor project where I did use a CO/ALR outlet to convert to copper, and I've not had any issues with it since. Still, I want to go back sometime and put Alumiconns there so I can get a proper GFCI in (no GFCI outlets are rated for aluminum).