Originally Posted by TimArchy

The only problem with a conversion is that they can easily start to push the $200 mark before you know it. Frame, $50 if you're lucky. Rear wheel, $75-100. And then maybe a new BB if not a new crank. hopefully you can save the rest.

Just be aware that, though it has been done, not all conversions are super cheap.
I've worked in a shop helping people convert for as cheap as possible (mostly helping them find parts). it worked about 1/3 of the time. The other 2/3 were better off getting a cheap track bike.

If you've got $350, let someone else figure it all out for you.