Helen's got it bad for Bertram and she'll do anything to land him as a husband. She chases/stalks Bertram to Paris, cures the king's mystery disease, and wins a chance to choose any husband she wants. (That would be Bertram, of course.)
Bertram isn't ready to settle down with a "poor physician's daughter," so he refuses to sleep with his wife. He tells Helen that he'll never be a husband to her unless she can meet the following conditions: (a) get the ring off his finger and (b) get pregnant with his child. Then he runs away to Italy and starts sleeping around with other women.
Helen wins Bertram back by tricking him into having sex/getting pregnant with his child and... you guessed it, getting the ring off his finger. Game over.