I used the awakening to my advantage. At first I would get up and clean for a bit...that usually got me tired enough to fall back asleep.
Once that stopped working (rather fast actually)....I started working out when I woke up like that. It took about two weeks, but by then I was so fatigued by working out that I slept through the night without a problem. deep...deep sleep. Of course then once I made it through the night without a workout, my day workouts become more intense.
Now that I look at it...I probably could have just made my day work outs more intense to begin with. Of course intense for me usually involved and hour of cardio and 45 minutes of weightlifting.....so pretty tiring.