The weir is on the River Avon just downstream from Pulteney Bridge (ST7564 : River Avon, Pulteney Bridge). It is shown on seventeenth century maps upstream of “The Monks Mill" on the west bank and "Bathewick Mill" on the opposite bank. The purpose of the weir would have been to provide a difference in river level (a 'head' of water) that would drive the water wheels used to power the mills.
In the early 1970's the weir was rebuilt in its current 'V' shape with an associated flood control gate (sluice) on the east side of the river (Link Bath Heritage).
The weird inscription on Kingston Buildings ST7564 : Kiŋston Bildiŋz, near Bath Abbey looks as if it was done by a proponent of the phonetic spelling campaign. This proposed the removal of all superfluous letters, such as the 'u' in 'buildings', and the introduction of the 'ŋ' symbol for the 'ng' sound which occurs in Germanic languages but not, to the best of my knowledge, in Romance or Slav languages.