I'm writing a Stellar application. I'm looking to use the API provided by stellard without running a copy of it myself.
Is there a public API server for Stellar?
Yep:
Stellar Foundation runs two instances of stellard you can use if you don't want to run your own.
network address RPC Port Websocket Port
test test.stellar.org 9002 9001
live live.stellar.org 9002 9001
If you want to try the new API look at this page: https://www.stellar.org/galaxy/getting-started/