Well, you know what the answer to your "What is one of your weaknesses?" question will be. The good news is, you have identified it and have 3.5 years of school (at least...unless you're a genius) to work on it.
So work on it...if you think it could hurt you in your career...it probably will (even if it's only cause you think it will).
Being introverted is tough to overcome, but toally doable. Like someone else said, being a TA could help. Try getting a part time job tutoring something too. Try to take on leadership roles in organizations...and I mean leadership roles that make you interact with people and lead people...not just secretary (taking minutes and emailing) or treasurer (cause a checkbook and spreadsheets don't talk back and have issues and personalities you need to learn to deal with)
It's not easy to do, but the only way to stop being introverted is to build confidence in your ability to lead people and communicate...and that happens, most often, through time and experience.
I too was pretty introverted at the start of college (10 years ago) in fact, one of my first professors (art professor) wrote me a letter of rec and she showed it to me and said in it "He's a shy modest person." I laughed because that's nothing like what I am anymore, but she was commenting about how I used to be. Another letter by someone who only knew me a year and a half kept talking about my loud, commanding nature (person only knew me as a student leader).
For me, a lot of things helped me get over the shy introverted nature. I played music with friends and played a lot of live shows on stages. I took on leadership roles in student, and eventually, national organizations. I tutored people. I became a teacher, etc. I made art and put it out there into shows and stuff (nothing like learning to take criticism from complete strangers to harden you).
And then, after doing a bunch of stuff to help overcome your problem...you have a great answer about weaknesses and how you worked on that weakness.