For Church of England records, you should contact the Durham County Record Office (now known as 'The Story'), which acts as the official repository for the Diocese of Durham and holds the parish registers for Jarrow. If you were baptized Catholic, you must instead contact the Archives Department of the Diocese of Hexham and Newcastle, as they maintain their own separate sacramental records. Since 1963 is relatively recent, if the specific register has not yet been deposited in these archives, you will need to contact the individual churches in Jarrow directly to check the physical books still held on-site.
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