ubiquitin: If he has remarried that would almost surely indicate that the (first) marriage is over and your conclusion is correct that he must give a Get. (I say almost because non-Ashkenazim may be different in this regard.)
But if he hasn’t remarried then I’m not clear what criteria or yardstick you’re using to classify that “the marriage is over”. It ain’t over until the fat lady sings; he still maintains the halachic right to challenge her request for a Get. And unless a Beis Din, after conducting a trial with the parties in attendance, rules that the specifics of the case fall into a halachic category where he must give a Get even if he doesn’t wish to (which is a rare scenario under Halacha), then the marriage isn’t over.