The poster removed the birth mothers name from the opening post, see replies #1 and 2. Her name is not unknown, it's her whereabouts that has been requested.
Rootschat isn't a forum for finding missing people. While I empathise with the OP and the quest, adoptions back in the 1960's were on the basis that the children would not find their birth parents and visa versa.
However with the change in attitudes over the years, there seem to be more people trying to find each other. However, the birth mother is this case may still not want to be found.
If the op hasn't done so already, I suggest registering here
https://www.gov.uk/adoption-records/the-adoption-contact-registerand hope the mother has also done so.
There are specialist agencies who work to find birth families, there are links at the bottom of this page
http://www.baaf.org.uk/It is highly likely that the lady in question subsequently married, so I would check the marriage index to see if you can follow through on the entries there by buying the marriage certificates to see who exactly who she married, see if she had anymore children and then try to find her in later electoral registers. The downside to this approach is that the maiden surname and combination of first name is not unusual although there are only 2 in Wales, in 1963 and 1964.
Hope this helps
Dawn