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The Third Edition of Pevsner, updated by James Bettley in 2015, says: "Immediately N of the church the Guildhall. Late C15, two-storeyed, timber-framed, and with upper oriels. Their sills are carved with foliage and a scene with a stag; but all this is thoroughly re-cut. Angle-post with the Archangel Gabriel. The exterior, was restored , the interior replanned and a new wing added in 1874-5 by J. K. Colling as master's residence for the grammar school."

The Grade I listing is
here and was last updated in 1998. It confirms late C15 and says it is the Gildhall of St Mary (the adjoining church is St Peter and St Paul).

The Guildhall is now a private house and was for sale a few years ago. The sales brochure referred to it as being in essentially the same form since 1475, and notes that it was later a Grammar School.

This Gildhall was pictured in the Suffolk Historic Buildings Group newsletter "Eavesdropper" number 41 (Winter 2010).
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Here are some detail pictures of the oriel window and the angle-post with the Archangel Gabriel.
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The PSIAH list of gilds (see References) shows two in Eye: The gild of St Mary Virgin and the gild of Saints Peter and Paul.
For information on buildings mentioned in the Pevsner guide to Suffolk in this location see pevsnersuffolk.co.uk