The first successful birth following transplantation of ovarian tissue has occurred. Stephanie Yarber, who experienced very early menopause, received ovarian tissue from her twin sister, Melanie Morgan and has given birth to a child.
You can read the BBC story.
Oddly enough, the oldest research article currently included in Pubmed describes a similar procedure carried out over a century ago:
The penultimate line from the abstract: Neither of the recipients had begun menstruating postsurgery at the time of this article's publication (about 6 months postoperatively), but signs were hopeful.
I wonder, well after the fact, if the women involved suffered any complications, given that this was well before the discovery of histocompatibility (and soon after Landsteiner's elucidation of blood types) and they were most likely unrelated.
It's also worth pointing out that this abstract promises a "review of literature." Success has been a long time coming.
(For the curious, the actual point of this kind of transplantation is not the extremely rare case of identical twins, but to let you remove ovarian tissue from a woman who is going to be subjected to systematic treatments such as chemotherapy, later returning it so she can be fertile again after a procedure that might otherwise leave her sterile.)