How to test the validity of your Sphinx documentation?

I have a whole bunch of documentation for my Python package written using standard Sphinx .rst files. I also have tests for my package, among which I'd like to include a test for whether the documentation will compile properly into the expected output. Basically, I want to catch cases when I've used a link to nowhere, or have a poorly-formed header etc.

Now I know that I can always write a test that calls make html and tests for the exit code, but this feels really dirty, so I'm thinking that there must be a better way. Anyone know what it is?