You should use the UPnP protocol to query your router for this information. Most importantly, this does not rely on an external service, which all the other answers to this question seem to suggest.
There's a Python library called miniupnp which can do this, see e.g. miniupnpc/testupnpigd.py.
pip install miniupnpc
Based on their example you should be able to do something like this:
import miniupnpc
u = miniupnpc.UPnP()
u.discoverdelay = 200
u.discover()
u.selectigd()
print('external ip address: {}'.format(u.externalipaddress()))