The large blocky building next to the church is the former abbey's 12th-century tithe barn, the place where people brought their obligatory contributions to the church, usually 10% of whatever they produced. This particular one has had a chequered history (it was a 17th-century theatre and a 20th-century disco, among other things) but it has now been fully restored and converted into a heritage centre and a food hall focusing on locally sourced Welsh products.
Pride of place goes to the Abergavenny Tapestry, produced by over 60 local volunteers over six years to mark the new millennium. Within its 8m width it depicts the history of the town; look for faint messages from the stitchers in the borders. Elsewhere on this floor a combination of artefacts and touch-screen monitors tells the story of the town and the abbey in an excellent interactive display.