Vodka. For the baby.
Seriously, the only thing we found effective was to sleep when the baby sleeps. Unfortunately that doesn't appear to be an option in your case.
For ours, we had the baby in our room for the first year. Unless baby has a need (food, changing), the baby can still hear you while you sleep and more easily comfort itself back to sleep. This works better when they're older, but there's going to be little benefit for a 3 week old.
*YOU CANNOT HOLD A BABY TOO MUCH* With that said, you do still need to put the baby down often enough to get your own tasks done.
Until very recently in human history, babies slept next to Mama for the first 2-3 YEARS and never was put down for more than a few seconds. A sling to help carry a baby while gathering food may have been the first human invention. Putting a baby down in a safe place wasn't an option because there was no safe place until 3 or 4 generations ago.
By about 3-4 months, the baby will start wanting to explore. Until then hold the baby all you can; you'll miss it soon enough.