Anne was first married to a Derbyshire gentleman, Christopher Horton (sometimes spelt Houghton) of Catton Hall, on 4 August 1765.

He later had the Royal Marriages Act 1772 passed to prevent any descendant of George II marrying without the consent of the sovereign, a law which remained in effect until the passage of the Succession to the Crown Act 2013, which, in addition to several other modifications, limited the requirement to obtain royal consent to only the first six persons in line to the throne (rather than all descendants).

As the Act's provisions could not be applied retroactively, Anne and the Duke's marriage was considered valid.

[5] While she was generally considered a great beauty, Walpole thought her merely "pretty", except for her green eyes, which he admitted were enchanting.

That her eyes were remarkably expressive is confirmed by the several portraits of Anne by Thomas Gainsborough, one of which is in the Hugh Lane collection.