Are you? Amazon is not known for posting profits, and is quite possibly operating AWS at a loss.
They have promised to show financial results for AWS itself with the quarterly report being released next month, so we'll see.
Update: in FY2014, they sold $89BN ($70.1BN in products, $18.9BN in services), and their cost of sales was $62.8BN. I am pretty sure that their margins are razor-thin in retail, but the service side had higher margins.
https://gigaom.com/2013/01/07/amazons-cloud-is-bigger-more-p...
They built it so they could run Amazon.com off the infrastructure, but then decided they could make money leasing their under-subscribed portions.
If they are not making a cut-and-dry profit from AWS (I'd fathom they are, they are one of the most expensive cloud providers and many other providers turn a healthy profit), then they are at least dramatically offsetting their own Amazon.com infrastructure costs which are mandatory anyway.
http://www.networkworld.com/article/2891297/cloud-computing/...
That doesn't mean the AWS part is in itself unprofitable. I'd wager it is, given how the business is somewhat mature and they have a huge chunk of that market.