In the first season, it is supposed to be a Gunnison and Griswold, but it's really just a brass-framed Colt. There's an excellent article in the current issue of Guns of the Old West that talks about the trials and tribulations of accurately representing firearms in the show. They knew that the gun was wrong, but it was the best that they could do given the time and availability constraints that they faced.
This season, he's got a Remington 1858.
Well we don't rent pigs and I figure it's better to say it right out front because a man that does like to rent pigs is... he's hard to stop - Gus McCrae