Have you ever watched a movie so many times that you can quote entire scenes without even thinking about it? Have you ever listened to a song and before you know it, realized you know every single lyric?
My best advice for memorizing anything is to recite it (or listen to others recite it) out loud. Writing your lines over and over would be a great idea if ultimately you were required to write them for the performance. But consider that performance is more audio than visual. Practice your lines (with other acting as the other characters in the scene) just as you would perform them. You'll be amazed as you realize certain verbal cues keep you going. Be sure to work on different inflections and emotions as well, because, again, these audio nuances will trigger your memory better than writing the lines over an over.
Start by reading your scene (out loud) a few times through. Then, slowly attempt to put down the script and see how much you can remember without looking. If there is a particular line you always seem to forget, add a gesture to that section to help remind you of the line. Keep it natural, of course, so that you can use it in your performance as well.
They say there is a direct link between music, motion, and memory. The lines themselves have a musical quality so don't feel you need to "sing" them in order to remember. However, I do encourage you to get up and move about, just as if you were on stage, as you practice. It may sound silly, but it really works.
The link below contains some other helpful ideas to consider.