Will this company hire me?

I applied for an internship at this large construction company in new york city, got an interview which went great and on the application it asks if I have ever been convicted of anything more than a minor traffic violation i checked yes and put down criminal mischief which is a misdemeanor. Two weeks later they called me back and said they wanted to hire me and they needed to do a criminal background check...this criminal mischief happened a year ago and I am still on accelerated rehab which is unsupervised probation and i cant leave the state without someone knowing, I live in Connecticut and the job is in New York. When the company does this background check and sees all this esp that this charge happened less than a year ago will they still hire me?