A major theme in much of Kate Chopin's writing is that of women feeling unhappy and repressed in marriage. So, Calixta's tryst with Alcee represented Calixta taking control of her happiness, through an affair with another man. After this incident, she was happy, laughing, and much more laid-back than her normal self.
If we look at the text, Alcee writes his wife, not to tell her of the affair, but to say that he missed her but was doing okay if she wanted to stay away a bit longer. This letter seems to indicate that Alcee is planning on continuing his affair with Calixta, since he is encouraging his wife to stay away on vacation. But, it probably isn't going to be a permanent thing; he has a wife, after all, and they must reunite at some point. So, from that evidence in the text, I would assume that they will probably hook up again, but it will be for them, something that they do occasionally when circumstances allow, without altering the main course of their lives.
Whether they will remain "happy" is an interesting thought--if we are to believe Kate Chopin's slant, that even Alcee's wife was relieved to have her "first free breath since her marriage" while she is away from her husband, we could assume that yes, this is the "solution" that all of them have been looking for. Alcee won't have a nagging or jealous wife, and Bobinot will be pleased to say goodbye to the "overscrupulous" Calixta that seems to have vanished after Alcee's visit. It will, most likely, last for some time. But, Alcee and Calixta might tire of each other after a bit...if they do, they will most likely find someone else to be with, in order to have that "outlet" that Chopin seems to find so empowering and necessary in order for married life to be survivable.
I hope that these thoughts help a bit; good luck!