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Telecoms giant Vodafone sparked an outcry when the proposal was revealed last month.
The clock, which dates back to 1910, was described as the nearest thing Morningside had to a town centre and locals were said to be “horrified” at the prospect of a sky-high mast positioned next to it, near the junction of Morningside Road with Cluny Gardens.
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One councillor said the mast would dominate the view and spoil an important social space.
Locals also pointed out the three cabinets to be sited along with the mast could also inhibit access to the manhole in the pavement which takes the community's Christmas tree every year, complete with power supply.
Now Vodafone has withdrawn the application and said it was considering two alternative sites nearby.
Morningside Labour councillor Mandy Watt, who took up the issue on behalf of all the ward councillors, welcomed the move.
She said: "I met them and urged them to find other viable sites and do whatever needed to be done to make possible sites viable."
She said one of the sites being looked at was near the Bank of Scotland car park on Belhaven Terrace and the other was where Cluny Avenue meets Hermitage Terrace, near the taxi rank. In the second location the pavement would probably have to be built out a bit.
Councillor Watt said either of the alternative sites would be more suitable than the original proposed. "They're away from the iconic landmark and if the one on the corner of Cluny Avenue and Hermitage Terrace possible that would be almost screened from view by trees at the railway and Hermitage Terrace gardens."
But she warned if the company decided against the other options it could submit a fresh application for the original site.
A spokeswoman for Cornerstone, who manage Vodafone’s masts, confirmed they were looking at the alternative sites.
She said: “Cornerstone always aims to work closely with local communities and planning authorities. We follow best practice principles in the location of our mobile infrastructure sites.
"A lot of consideration, thought and time goes into proposing where our infrastructure should be placed. Our aim is to minimise local impact while addressing practical technical constraints to enable quality digital connectivity to the communities we serve.
"We are aware of some concerns regarding the location of the proposed telecommunications site at Morningside. After further consultation with local stakeholders we are now exploring the viability of locating a base station at two alternative locations, close to the original proposed site.”
Lothian Tory MSP Jeremy Balfour said he was delighted to hear the application had been withdrawn and called it “a great win for the local area”