The centre holds records of the Catholic Church in Scotland dating back to 1177 AD.
He is due to be sentenced next month following background reports.
Custodial sentence
The SCA looks after administrative, financial and legal documents of the church and many items are letters sent to churchmen from overseas and are valuable primary sources from the early days of the British Empire and the colonisation of America.
Archivists only realised he had stolen from them when Fallon was arrested for a similar crime in London and told police he had done the same in Edinburgh.
They discovered 288 files were either missing or damaged.
The court was told visitors who want to view archives at the SCA must be interviewed by an archivist and are only allowed to take a pencil and paper into the reading room where they are shown the documents, but are not allowed to remove them from the building.
Sheriff Noel McPartlin warned Fallon he could face a custodial sentence.