Interesting, and this is sort of the mirror opposite to my position. While I'm probably a slow memorizer in comparison to the people here, my teacher reprimands me for reading the page a few times, and then playing it by memory, basically memorizing "too quick".
Anyway, some thoughts:
Take up a new piece of music which you have not read before, and approach it with the mentality that you will memorize it from the beginning. That is, you need to wean yourself off the page as soon as possible. And I'm not talking days or weeks, what you should try to attempt is to play a few measures and memorize them immediately, within a few minutes. For that, you will need to try to get yourself to observe the patterns going on in the music and put them into some kind of structure which helps out with memory. (For professionals, this structure may well be subconscious, but it's there nevertheless imo.) Observe chords and harmonies, repeats, transposed elements, etc. which can help you memorize the music quicker because it reduces the amount of stuff you need to explicitly memorize. Figure out the melody if any and play it individually/hum it out loud. And so on. But the key is to have a certain amount of time for memorization where you are allowed to use the page, and then you should try to recreate it from memory. Give yourself a certain amount of time -- I prefer to do five minutes. Try to memorize 2 measures, or 4 measures, depending on the density of the music. Then, chuck the music aside (to another room if you have to) and play it from memory. Then, try to recall the bars of music after 8 hours, the next day, etc.
I think this also eventually primes you to memorize music more readily.