The blaze took hold of the Royal Victoria Hospital building at about 16:20.
Firefighters used six pumping appliances and breathing apparatus to battle the blaze. They were still at the scene a number of hours later.
A Scottish Fire and Rescue spokeswoman said there were no casualties in the incident.
The fire was reported by a significant number of people as smoke rose from the abandoned building.
The Royal Victoria Hospital was closed in 2012, but later temporarily reopened to deal with seasonal demand.

Llanbedr airfield, near Harlech, is one of eight which has been looked at.
Gwynedd Council says it would support the development, but the Snowdonia Society is opposed to the impact it could have on the wider national park.
The views of local people will be sought if the site is identified as a serious contender for the role.
By the year 2030 the global space economy is expected to be worth Â£400bn a year and the UK government wants to secure a slice worth Â£40bn annually.
But it needs a port from where it is thought satellites and space tourism flights will be launched, prompting the consultation.
Six of the locations under consideration are in Scotland and one in England.
Gwynedd council leader Dyfed Edwards said if realised the plans have the potential to create "significant economic development and the growth of both the local and national economy" at Llanbedr Aviation Centre and Enterprise Park.
"As a council, we are delighted that the Llanbedr airfield site has been selected as one of eight possible locations for the UK spaceport," he said.
"And we have written to the UK government to confirm that we would fully support the development of the site as the location of the UK spaceport.
"The Llanbedr site has a great many attributes which would make it an excellent choice, from the fact that it would be possible to use an existing airfield facility in a secure costal location where a spaceport could operate safely, to its close proximity to both Aberystwyth and Bangor universities, both of which have expertise in fields relevant to space technology."
He said the authority would continue to work with the airfield owners and the Welsh government to exploit the site's potential.
But not everyone agrees with the proposed location, including some local residents and the Snowdonia Society - a charity set up to "protect, enhance and celebrate Snowdonia, its wildlife and heritage".
Society director John Harold said: "Whilst we support and want to see appropriate development in the park, we feel that something like a spaceport would bring levels of impact on wildlife, on noise and disturbance and other issues that would not be appropriate for the national park.
"We already have a world class facility here. We already have a world famous brand. It is called Snowdonia national park and it is a real success."
The UK Department for Transport expects to the publish a response to its consultation later in the year.

The SDLP and Alliance Party have yet to say whether they will join the executive.
Mr McGuinness said he was still hopeful both would be part of the new executive, but added it was now "make your mind up time" for both parties.
The Ulster Unionists have already said they will enter opposition.
The Alliance Party's ruling council is due to meet on Thursday to decide if the party will join the executive.
The majority of the posts in the executive will be filled by ministers from the DUP and Sinn FÃ©in.
In 5 May's election, the DUP maintained the 38 seats it held in the last assembly, while Sinn FÃ©in lost one and now holds 28.
The Ulster Unionists have 16 seats, while the SDLP have 12.
The Alliance Party secured eight seats during the election, meaning it does not have enough seats to automatically qualify for a ministerial department. However, it has been offered the justice ministry.
"We have offered the position of justice minister to the Alliance Party," Mr McGuinness said on Wednesday.
"They have come back with a number of issues which they would like to see addressed and we will respond to those during the course of today."
He added: "I think the SDLP have to consider what is good for the people, what is good for the role of the government as the Good Friday Agreement was all about inclusive administration.
"It's make your mind up time over the course of the next seven days.
"I hope the SDLP will decide to come into the administration, we are involved in discussions with them, and there will be a further engagement tomorrow and after that they will have to decide."
A fire at a disused hospital in Edinburgh led to thick smoke being seen across the city.
[NEXT_CONCEPT]
A four-month consultation which could help decide the location of the UK's first spaceport ends on Monday with one site in Gwynedd being considered.
[NEXT_CONCEPT]
The ministers of the new Northern Ireland Executive will be appointed next Wednesday, Deputy First Minister Martin McGuinness has said.