I used one of the various android Sleep Timer apps for a while (I don't really need to get up so it was just a test).

What they do is listen to your sleeping and wake you up between cycles. You set it for the last possible minute, and for the first time you are willing to wake up, and when you are already rousing out of deeper sleep, during that window, it turns on the alarm sound.

It worked pretty well. I immediately noticed that I never was startled awake or dragged out of a dream. I never had any of that period of wondering what day it was or why I had set an alarm.