I'm going to marry my boyfriend on December 15 2012
We've been going out for more that 1 year we know each other for more than 3 years.
I'm recently pregnant.

And my question is "can I give him papers after we get married"? Does henbane to leave the state? How much time does the process take? And how much does it cost?

He arrived here in the year 2006 and he has never had problems with the police or any any felonies.

NO... if you are talking A green card... you can't do anything... in fact him getting married only complicates the issue... the4 INS.. frowns upon these attempts by people thinking it entitles them to a green card when it doesn't. In fact they will take a lot longer because of that, IF he ever gets it... and he is no more entitled to it than anyone that is following the legal process to get here... if he's illegal now... then it only guarantees many more problems.

It doesn't matter when he got here... if he's not here legally NOW ( you didn't say if he has a legal status here or not and what it is)... they will know he is illegal... where he is and where to find him for deportation.