The dovecote wasn't actually part of the adjacent 13th Century priory, having been built in about 1600 by the Bulkeley family, well after the monastery's Dissolution. The building accommodated 1,000 nest boxes accessed by the birds via an aperture at the top of the dome and by humans (for pigeon eggs & meat) via a revolving ladder attached to a central pillar which I couldn't resist climbing; it's quite impressive inside.

You may (or may not...) recognise the dovecote from 'The Fever', a 2004 film starring Vanessa Redgrave and Angelina Jolie which doesn't seem to have been released in the UK. The domed building represented a disused Eastern European church.