Her distraught husband raised the alarm after she fell at the popular beauty spot.
A Stornoway Coastguard search and rescue helicopter was launched at 4pm on Saturday and recovered her body from the cliffs below.
Police said that despite the best efforts of those involved the woman died at the scene.
German-born Dr Weir, who lived in the Liberton area of the city, was a customer service worker. Her death is believed to have been an accident.
“Police have carried out enquiries and there are no suspicious circumstances. A report will be submitted to the Procurator Fiscal,” said a spokeswoman.
Neist is the most westerly point on Skye and is renowned for its rock formations, which are akin to those at the Giant’s Causeway in Northern Ireland.