Another option is to use Aptible: https://www.aptible.com/

It's basically (a better) Heroku with an emphasis on enabling HIPAA/HITECH compliance. They do both app and DB hosting (including Postgres). And it's on AWS so integrates easily with existing infrastructure/code.

Disclaimer: Biased as I know the founders and we use the product. But they are good people and it's a good product!

But aptible doesn't have managed postgres. I still have to do all the work of setting up a database and managing backups, upgrades, etc. The only thing aptible brings to the table is single tenant hardware and encryption by default. I can just get that from Amazon. Am I missing something about aptible? I don't understand the attraction although it does look fancy

Healthcare Blocks is another option: https://www.healthcareblocks.com

Aptible absolutely has managed postgres, and it's great. I'm using it at this very second.

Where can I find out more? All I see on their marketing site is "database containers" which I interpreted as little more than a container that happens to run a database, not as something that manages backups, point-in-time restoration, multi-az failover, upgrades, and all the other things that RDS has built in that many folks would otherwise prefer to have an (expensive) postgres specialist setup and manage for you.

Dang, sorry about that: https://www.aptible.com/enclave/

That page needs some love, thank you for the solid comment. I think the features we list cover your comments, but let me know if you have specific questions or other features you'd like to see!

Source: am Aptible CEO

Although dirty, you could still technically use Postgres I believe, you would just need to run it on EC2 and EBS instead. The covered services for the BAA include EC2 and EBS, so you would still be covered under the signed BAA with Amazon. When it is eventually covered under the BAA, you could port over to Postgres RDS.