Hi greentea,
Not that I myself am qualified to give careers advice but here's my thought for what it's worth.
I know lots of people in the web development industry that have gone into it (and make a good living out of it) by extending what was no more than I hobby. Sometimes they have backgrounds in design but often they have backgrounds in general programming or even business consultancy.
Moreover, the barriers to entry into web development are pretty low so as to be irrelevent; all that counts is your personal skillset that you can bring to the job. Abilities such as being able to interpret "customer" needs (whether that's a paying customer or your own marketing department) are real assets ... and you're not necessarily going to learn that in *any* degree course!
As someone who has employed web developers (note: developers, not designers!) in the past I have seen CS BA students command a higher salary but they often failed at front line work with clients.
I hope that helps a little. Good luck with your career and keep visiting to let us all know how you're getting on.
Regards,
David