I've extracted the link for you, but beware that it is a huge file (790mb)! It will take some time to download. I did it a while ago, but it was not easily repeatable for other sites so I'll just give you the link here. (Even though flashgot did NOT work this time, it does often work for downloading flash videos, so that's quite a good firefox add on to use in general.)
How I did it in this case (may NOT work for other flash streams):
You need to have the noscript addon for firefox installed for the following to work:
- With no script blocking all js and flash (as default for unvisited sites), go to the site- e.g. http://liveweb.arte.tv/fr/video/Gonzalo_Bergara_Quartet_Festival_Django_Reihnardt/
- Now mouse over the blocked flash plugin, which gives away the server they are using, which is
download.liveweb.arte.tv
in this case
- With this knowledge, go to google and do
site:download.liveweb.arte.tv jazz
or another search term.
- Then when you find the link you want, go to the google cache page instead of the actual live page.
- Then, make sure arte.tv and googleusercontent are allowed in the drop down noscript menu, but everything else is blocked, and you should be able to mouse hover over the disabled video plugin and copy the link.
Results may vary for different videos from the same page (http://www.my-jazzlive.tv for example). But is possible that when reaching the cache page, more usual ways of downloading may become available, like Flashgot: so keep an eye on Flashgot Media button!

Whether this will work for other sites is anyone's guess!