Builder first removed walls from digs, then registered costly gazebo (9)

I believe the answer is:

belvedere

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'gazebo' is the definition.
(belvedere is a kind of gazebo)

I cannot understand how the rest of the clue works.

(Other definitions for belvedere that I've seen before include "Midlands Big House and prestigious north Dublin school" , "Summerhouse with view" , "Pavilion with a view" , "Mullingar house and north Dublin school" , "Gazebo - lantern" .)

I've seen this clue in the Sydney Morning Herald.

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