REM periods occur throughout the sleep cycle, with the first one usually beginning around 90 minutes after you fall asleep; there is not only one REM period.

That said, though, REM periods do occur more closely together as your night's sleep progresses, so you do indeed have a much better chance of catching REM after 5 or so hours' sleep... so you will tend to find REM quite easily after 7 hours' sleep.
My bad for terrible wording. I mean I notice my dreams later on in the night, quite easy actually. I don't remember dreams from my first REM though, I think it's because dreams only last upto 10 minutes.