The sad fact is, almost no one gets a game design job immediately on graduation. Most people enter the industry as a programmer, an artist, or a software tester. The reason for this, I think, is that those positions are inherently low risk: a programmer or artist who screws up their entry-level work is probably not going to sink the entire multimillion-dollar project; another programmer or artist will clean up the mess and no one else will notice. (A tester, by definition, can at worst do nothing.)
But a designer's job is, essentially, to tell the programmers and artists what to do. Design mistakes don't just require another designer to make repairs, but also programmers and artists to redo their work. Design mistakes bleed across departments, so companies tend to be very careful of who they hire in those positions.
As such, anyone interested in game design should also minor in Computer Science, Art, Business, Marketing, or some other discipline that is directly related to game development. This gives you three ways to enter the industry: game design (unlikely), QA (easier to find, but tedious and boring to most people), and whatever you minored in. Understanding multiple disciplines also makes you marginally more attractive to a game company (especially a small one), since you can fill more than just one role and you're more likely to communicate well with people in different departments.
Additionally, take a second minor in something that has nothing at all to do with games. Game designers need to foster a lifelong passion for learning (just look at all the crazy stuff that Will Wright or Sid Meier knows – the stuff that has nothing to do with games – like astrobiology or world history). I know, I know… most educational institutions do their best to squash the love of learning out of every last student. That makes it all the more important to reverse the trend in college, by studying in-depth something that really interests you. It could be anything; astronomy, art history, classics, Shakespeare, French, quantum physics, whatever. There are many reasons why this will help you. First, it will set you apart from others by giving you unique interests. Second, it demonstrates your all-important love of learning. Third, it gives you a small, random chance to be a perfect fit on any given dev team (example: if you minored in abnormal psych and unbeknownst to you, the company you’re applying to happens to be in negotiations on a game with a paranoid schizophrenic as the main character…).
Two minors? Sounds crazy, but in a field with practically no entry-level positions open to new graduates I consider it a necessity.