After reading the other posts, which where good ideas for a temporary fix and will work great but personally think the underlying problem is a hormone imbalance. After starting on the pill I am guessing that your girlfriend is
having a major fluctuation of hormones resulting in the
growth of facial hair. Many women have this happen during
pregnancy etc. You should read up first on the dangers
of the pill and the havoc it can do to the female body and
then read Dr. John R. Lee's writings on hormones. I hope this helps.