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With government approval for the new hospital next to the Royal Infirmary at Little France still awaited, Mr Briggs said it was clear the existing Eye Pavilion on Chalmers Street would have to remain in use for a few years yet.
But he said problems with the building, including lifts which frequently break down, meant a funding commitment from the government was needed.
The existing building was opened as a state-of-the-art hospital in 1969, but was declared not fit for purpose in 2014. Among the problems are roof leaks and lifts which frequently break down.
A new hospital was approved in principle by the Scottish Government in 2018, but in December 2020 ministers told NHS Lothian they would not fund the project.
After a public outcry, Nicola Sturgeon promised during the election funding would be available and the new hospital is now included in the recently-published NS Recovery Plan.
Mr Briggs said: "I’m pleased we have seen progress, with the government U-turning over the need for a new hospital, but there are long-standing issues with the fabric and maintenance of the current building and they will need to be addressed because there's not going to be a new hospital built overnight.
"For the past year people have been emailing complaints that lifts are not working. Until we get the new hospital we need to see funding to maintain and potentially refurbish the old Eye Pavilion.
"Most people who use that building are seeing that need – it’s in a poor state.
"NHS Lothian is the worst funded health board in Scotland, so it doesn't have pots of money sitting ready for that. The government needs to be committing not just to a new hospital, but to making sure the old one is fit for purpose for patients and staff."
Sylvia Paton, who has attended the Eye Pavilion regularly ever since it opened and is a leading figure in the Keep Edinburgh Eye Pavilion campaign (KEEP), agreed the current building was in a poor state, but was wary of spending too much money on it.
"They need to do essential repairs rather than tarting it up,” she said.
“The lift is absolutely essential – I don't know whether that means a whole new lift system or a new motor or refurbishing the existing motor.
“But I don't think anything beyond that is good value for money.”
Jim Crombie, NHS Lothian deputy chief executive, said improvement work had recently been undertaken at the Eye Pavilion, which included essential repairs to the lifts and the roof, with work continuing to maintain the building.
"We are continuing to ensure that patients receive their treatment in a space which is both comfortable and welcoming,” he said.
“Despite the building’s challenges, our focus and commitment remains on delivering safe, effective and efficient patient care.
"NHS Lothian will continue to support this building’s infrastructure and accommodations until we open our brand new eye hospital.”
A government spokesperson said: “The Scottish Government is committed to the replacement of the Princess Alexandra Eye Pavilion, and we will be setting out the next steps in the process shortly.”