Well, my degree was a Bachelor of Science in Computer Science.
In terms of going after a position in design, there are a few things I would personally suggest keeping in mind while going through school...
The degree helps for several reasons -
1)It's a degree. It shows you can put in a lot of hard work and finish something that requires years of dedication.
2)A good computer science degree will teach you not only how to code, but how to prototype and design good systems.
3)It's hard to find good programmers willing to work on your ideas, and this is the reason I did computer science myself. I wanted to design games, and I'd program them myself if I had to.
4)You can talk to programmers and understand technical limitations.
However don't just go to College/University just for the degree (or the whole 'college experience', although you certainly should get into that as well), there are a few other things you should make sure you do while there.
-Work on your own projects. Most Computer Science departments in this day and age have at least one class or outside organization dealing with computer game development. Become involved in this (and if it doesn't exist, I assure you that there will be at least enough of your classmates interested to allow you to get one organized yourself). A finished project is 10 times as valuable as an unfinished one, but you'll learn something from them all, completed or not.
-If you want to be a designer, don't just be a tech head. Take some English, History, Communications. Broaden your horizons, make sure you can write and convey your thoughts in a clear and concise manner. People are going to need to understand you, and you're going to need to have interesting things to say, these classes can help with inspiration and then infecting others with that inspiration.
-Play games! Make sure you don't bury yourself in work and forget about exactly what you're working towards. This bullet point is at least as important as anything else you will do, if not more so. Play board games, play pen and paper games, play computer games, and think about them. Working in design you will hear phrases like, "Yeah, this encounter is totally like the third boss in the dark world of A Link To The Past
" (note: this is a hypothetical example, don't actually look for a Moth + Deadly Spikes encounter in Vanguard, it may or may not be there).
Anyway, I hope that helps some. If you have any more specific questions or the like, feel free to PM me, and I'll do what I can to check back in on this thread as well. Good luck with the whole college thing, and above all else, enjoy it while you can. You'll need those memories of the sun to cling to once you start working on your first game.