Interesting to read other people's take on this building. For those not familiar it sits over the south side of the Roman Amphitheatre just west of St John's Church. Because it is listed Grade II, but has lain empty for 25 years, and needs a lot done to it, it has been very difficult to get anyone to take it on. They are trying to get someone this year. As has been pointed out it was the Convent School founded here in 1854.
The belief that the central portion was built in 1730 depends on de Lavaux's plan of Chester in 1747 which has written down the outside of the east boundary of the plot "Mr Comberback". James Comberbach in his will in 1736 left two new built houses in or near St John Street near to St John's Church, which had tenants, and the large house in which he lived on St John Street. However these are not Dee House. His nephew might be the Mr Comberback of 1747, but his will also left a house on the west side of St John Street, which later became the Mechanics Institute, the City Library and was rebuilt as "Cruise" nightclub.
So who did build it? I ask because I'm trying to find out, so some readers may know something. It certainly means going back before 1854 - there aren't any answers in books or on websites, other than speculations already quoted. It is likely to be part of the land of the Bishop of Chester that was leased to Sir Robert Cunliffe (son of Foster Cunliffe the 18th century shipping magnate in Liverpool) and to his son Sir Foster Cunliffe, until the latter bought Acton Park near Wrexham in 1783. It was afterwards let, and when advertised for sale in 1831 was a sizeable building (the drawing room alone was longer than the central section of the Convent School, and it was more like 1760s in date.
In 1858 the Convent School erected a new building, and in 1866 Edmund Kirby, the Liverpool architect, designed the east wing with its distinctive brickwork, and the chapel mentioned by others. At the time there was no sign of the building claimed now to be a Georgian House. It is possible that Kirby connected the new building to the old re-using materials from the old.
So why make a fuss? Well a lot of the pressure to restore it exactly as it is stems from claiming it is based on a Georgian mansion. If it isn't, and is merely fragments of one neatly incorporated in 1860s building work, it wouldn't be so crucial to restore it to such an extent, and something more affordable and more attractive could be attempted. There is little use demolishing it to recover the amphitheatre as cellars are recorded, and the chapel block foundations were dug down ten feet. But it might fair better if a less exacting restoration were possible and it could be made into something usefully complimentary to Chester.