That's an oversimplification - menopause results from decreased levels of hormones that would normally be secreted during the process of egg maturation, among other potential causes [see ref 1].

Further, all the eggs a woman will ever have are created while she is developing in her own mother. This is totally distinct from stem cells, many of which are able to generate new stem cells throughout our lives, albeit at an ever-decreasing rate.

[1] http://www.mayoclinic.org/diseases-conditions/menopause/basi...

I dug into this a bit. A good metric by which to measure exciting (and very controversial) papers like the one referenced by that link is to see whether the papers have been replicated by others in well-regarded journals.

While the principal author of that article published an equally-controversial paper in 2005, follow-up studies suggest that there are no newly-generated egg cells in mammals [1]. I think it's fair to say, again, that stem cells don't play a role in maturation of adult egg cells.

http://www.ncbi.nlm.nih.gov/pubmed/23630252

Everything when it comes to reproduction is an oversimplification, but the basic problem is the same a lack of active stem cells (there are still stem cells present). Once all the eggs are used up then the ovary stops producing oestrogen and the other hormones.