I learn (and memorize) new songs (and sometimes new arrangements or keys of songs I've done before) every week for church gigs. While I work with charts, my process would work the same regardless of what the documentation is you're starting with.
1) Make sure what you've got is good. Run through it a few times to make sure you've got the part right.
2) As soon as you can, start looking away from the paper - a phrase here or there, where you know it, then half the chorus, the the other half, then the verse, finally the bridge, intros, etc. - the goal is to keep running through the piece, and withdraw the need for the paper, but keep (and polish) the performance. In my practice sessions a couple days at an hour a two a piece, and I've got the piece memorized, the rest of the week is polishing; sometimes adding little nuances that I didn't hear the first few times through.
The more you do this, the better you get at it. If you're playing a lot of stuff from any one genre, you'll find a lot of things that a lot of the songs have in common, and that speeds things up considerably.
If I have a week with a lot of tough music in it, I'll be listening to the music going to and from work in the car (my commute is an hour each way). That helps internalize things as well, especially if you're visualizing yourself playing the notes while listening.