Transportation from Basque area to Paris

Take a flight. The fastest trains will take more than 9 hours total, until 2015 at least when a new High Speed Line to Boredeaux will be completed.

According to the timetables at bahn.de, the fastest train route requires two connections and takes 12hr18min. According to the search engine skyscanner.com, the cheapest flights would be on Vueling, a budget airline. You would connect in Barcelona and arrive at Orly.

You could also travel from Bilbao to Biarritz (france) by bus (2-3 hours) and then take a direct flight to Paris, some flights are cheap like 80$.