I do remember this stage with my first daughter - some babies just don't seem to want to sleep and people didn't seem to believe me about how bad she was. With her, I got a sling and she used to sleep in the day in that, being carried around all the time. At night, she did learn to sleep in her basket, but I had to put her down asleep and she'd wake up more often than not. It does get better though, I promise!
There are some good suggestions to try above. Have you tried him sleeping on his side, you can do this propped up by rolled up blankets? Some babies just don't like sleeping on their backs when little.
I hesitate to say this, but will he sleep on his front? If that's the only way he'll sleep, and it preserves your own sanity, then I wouldn't rule it out. There is a higher risk of SIDS, but I'd suggest trying it while watching him, and if it works, you could get one of those baby sensing mats to go under him for your peace of mind.
Also, what's the mattress like? My niece wouldn't sleep in her moses basket until my sister got a new, thicker mattress, that did the trick!
I hope this helps, good luck!