Please don't make that baby cry himself to sleep. All he wants is the closeness to his Mommy like he had in your tummy. We "teach" our babies trust by letting them know they can count on us when they need us. He has no experience in this world, and it's our job to show him love and care by tending to his needs.
Crying is the only form of communication for a baby and it's up to us to figure out what they're trying to say. At 5 weeks old, he's crying because he likes the closeness of his Mommy to make him feel safe in this big, scary world.
To give such a young baby separation anxiety would be heartbreaking. Try the "Ultimate Sleep Positioner" at Babies R Us and put him in your bed next to you. It cradles him and has a built heart beat device to put him to sleep. Sleep close enough so he can smell your scent (I know that sounds funny, but that scent is so comforting to him).
I know it's hard to go without sleep, but the safer you make him feel now, the safer he will feel in the near future without having to be "attached". Once he knows he can trust that you will be there, the anxiety slows down.