Hi - Ok - an adopted child where there is proof that the birth mother is
Jewish is definitely Jewish according to halachah. The child whose
birth mother is not known for sure to be Jewish must undergo conversion
(bris, mikveh) to be halachically Jewish.
After conversion, the adoptee is regarded as Jewish and of course the
proper restrictions - i.e. not marrying Kohen apply. However, the
children are then born Jewish (if the adoptee is female).
Shoshie