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SORRY. YOU CAN'T BUY A BIRTH CERTIFICATE FOR SOMEONE BORN BEFORE 1837 ! ! ! !
Birth certification began in 1837 and was not compulsary for, I believe, a couple of decades.  I am sure you have already tried the IGI, but a word to the wise.  I have found that you need to use each individual search rather than the "Global" search as you get better results.  Even better if you can find out the Batch N°'s for Rothbury you are even more likely to get a result.  Good luck.

P.s.  I have family in this area at this time so if I come across your family in my searching I will post it here.