I can confirm that it is in fact accepted to be absolutely true, having been informed of it in a lecture at uni

I can see my job's more responsible than I thought!

I checked this in the OED & found quite a complicated story. The French 'gigue' (the dance) does seem to derive from the English 'jig'. But nobody seems to know whether 'jig' derived from the older French 'gigue' (the musical instrument a bit like a viol), and if so when or how.