Familiarize yourself with the content. Read the script through one time. This establishes subconscious visual imagery that creates meaning and associations with the words. Then, go back to the beginning and read through to a natural break. This will be the first segment you will memorize. Breaking the script into segments will turn the memorization into doable sections. (Segments should be at least several paragraphs in a monologue or several pages in dialogue.)
Walk around while reading, read the script out loud, try different voices, gesticulate, and always read your cue lines. Although you don't need to memorize your fellow actors' parts, you need to be familiar enough with them so they can serve as prompts for you, or help you recover if someone else drops his line. (In fact, if you follow these tips you might find that, inadvertently, you memorized the other lines.) Also, don't forget to visualize what you are saying: See the scene, your place in it and the other characters with whom you are interacting.
Start at the beginning by covering up the script with your hand or another piece of paper. Better yet, leave the script upside-down but nearby. Every time you make a mistake or get stuck, check your lines then go back to the beginning of the segment---not just the beginning of the paragraph or line.
Once you have memorized a segment, move onto the next. Every so often, when making a mistake, start from the beginning of the script so you have to move from one segment to the next. Soon, you will find you have memorized the whole script---in most cases more rapidly than you thought. When you are fairly confident in your memorization of the script, ask a friend to test you by reading the other parts. This will alert you to any dropped lines you may not have noticed and help you iron out any errors.