This is a really common problem at this age – so common it would more properly be called a "pattern" than a problem – but of course it really eats into your need for sleep, and that can begin to feel impossible after only a week or so.
At six weeks, your little isn't going to recognize a sleep routine yet, because she's still going through the massive developmental changes that will eventually bring her to where she CAN recognize patterns. So please don't just leave her to cry – the staunchest advocates of Cry It Out don't recommend trying that before six months.
Dr. Harvey Karp, author of The Happiest Baby on the Block, explains in these short videos why he calls the first 3 months the Fourth Trimester, and explains why babies run pretty much according to their own needs for the first 3-4 months:
How-to's: http://www.youtube.com/watch?v=G6KnVPUdEgQ&feature=re...
Enhancing sleep: http://www.youtube.com/watch?v=Tk5MUOMecHI&NR=1
Interview: http://www.youtube.com/watch?v=iu0TtxO-ocY&feature=re...
One thing I don't think Dr. Karp mentions is that the brain's day/night pattern, which is partly governed by the body's chemicals, is affected by daylight. Make sure your daughter is getting good exposure, especially early in the day, to natural daylight, which will help her brain produce the sleep-inducing hormone melatonin at night. Even during naps, the brain "registers" incoming light through the closed eyelids, so if daylight doesn't keep her from napping, try putting her near a big window with open blinds while she sleep.
Likewise, keep lights always low in late evening and through the night. If you need to attend to her during the hours of darkness, use as little light as possible, and keep it out of her face.
Unless she has any high-risk factors for SIDS, belly-sleeping will probably not be too risky. When my daughter was born in 1971, belly-sleeping was not even thought to be a risk, and virtually every baby I knew was allowed to sleep face-down. I have never personally known a family who lost a child that way.
Do check out what Dr. Karp has to say about infant sleep. I know lots of parents who have gotten good results using that advice.
Congrats on your little one!