Can I be fired for having a criminal record?

I'm from North Carolina and I received the same form that NOT SURE did(mentioned in a previous post). At the top of the form the circle next to criminal record is check marked and all other circles (financial,driving record etc.) are not. So, I beleive that my company is doing a background check on my criminal record. Even though at the bottom of the form it states that they (the company) has permission to do a background ground check on the other items as well. Hmmmmmmmm
I have been working for a contract firm for over six years now and have a excellent work history with the company. When I first signed on with them I didn't have to sign any papers saying that they could check my background or even if I had a criminal record for that matter. Now I have to sign this piece of paper giving them permission to check my life history out. They say they must do this because the company that my firm has hired me out to is requesting this information from all of its employees. If you haven't guessed by now, I have a felony conviction, dated back to 1979 for robbery in the state of Texas. It is somewhat all in the past to me except for when companies etc. want to pry into my personal bussiness.
The question is, can my contract firm or the company that I work for fire me for having this convictions? I haven't lied about anything and no one has a problem with my job abilities etc.

I worked for a large company in Personnel for a number of years and I can only speak from my experience. It used to kill me when some nice person came in, applied and went through all of the hiring process and training and when we got his background check back and matched it with his application, we had to fire him for lying on his application. You see, it is the falsification of the application that got them fired, not the criminal history. I wish I had better news for you. If they want a background check, you signed your application and will get one, and dollars to donuts, you won't be working there no matter how old the felony is. We hired this great guy and he was the most perfect employee any one would want to have, and he also had a large family. He screwed up when he was 19 and it followed him. He was 30 at the time of hire. I got the job of letting him go. There are people who turn their lives around and some that don't. Unfortunately there aren't forgiving businesses out there that take that into consideration. You have a feeling it will come out and it probably will. Technology is faster now so I would look elsewhere and not worry about getting fired. Take care.