A degree might improve your chances with some employers, but it's not necessary. The main thing you should aim to get is a wide range of experience; you're more likely to get a decent programming job if you can demonstrate that you know a variety of technologies and are easily adaptable.

One year's worth of experience is not a lot. Doing a 3 or 4 year degree might help if only to get more experience under your belt. Otherwise, you might be able to find a job that doesn't require a large amount of experience, and use that to start your programming career.

Whichever option you choose: good luck!