Yes it is possible, and easy, it will (mostly) be in IPython core for 2.0, I would suggest looking at those examples for now.
[edit]
$ jupyter nbconvert --to notebook --execute original.ipynb --output=new.ipynb
It is now in Jupyter NbConvert. NbConvert comes with a bunch of Preprocessors that are disabled by default, two of them (ClearOutputPreprocessor and ExecutePreprocessor) are of interest. You can either enabled them in your (local|global) config file(s) via c.<PreprocessorName>.enabled=True (Uppercase that’s python), or on the command line with --ExecutePreprocessor.enabled=True keep the rest of the command as usual.
The --ExecutePreprocessor.enabled=True has convenient --execute alias that can be used on recent version of NbConvert. It can be combine with --inplace if desired
For example, convert to html after running the notebook headless :
$ jupyter nbconvert --to=html --execute RunMe.ipynb
converting to PDF after stripping outputs
$ ipython nbconvert --to=pdf --ClearOutputPreprocessor.enabled=True RunMe.ipynb
This (of course) does work with non-python kernels by spawning a <insert-your-language-here> kernel, if you set --profile=<your fav profile>. The conversion can be really long as it needs to rerun the notebook. You can do notebook to notebook conversion with the --to=notebook option.
There are various other options (timeout, allow errors, …) that might need to be set/unset depending on use case. See documentation and of course jupyter nbconvert --help, --help-all, or nbconvert online documentation for more information.