My wife and I have plenty of experience with this. You have to carry your current passport, as well as the old one with visa in it. Tie them together with a rubber band, because if the visa is for an infrequently visited country, you just might forget the old passport at home.

You could, of course, visit the consulate, pay a fee and have the visa moved to your new passport. We never felt like bothering, but that is up to you.