The cable, one of 14 which supports the newest bridge over the River Clyde, came crashing down on Monday night.
A Glasgow Council spokesman said public safety was the "number one priority".
An initial inspection of the Arc - better known as the "Squinty Bridge" - has been carried out to establish what caused the support cable to fail.
A further inspection by designers Halcrow and civil engineering contractor Edmund Nuttall Ltd, who built the bridge, is due to be carried out later on Tuesday afternoon.
It is hoped engineers will then have a better idea of the full extent of the damage.
"Until we are completely satisfied the bridge is safe to use, it will remain closed"
A spokesman for Glasgow City Council said temporary traffic diversions were in place to minimise disruption to the travelling public.
He added: "We don't believe the integrity of the bridge is affected. The Clyde Arc is designed to allow for the removal of one of the bridge supports at a time for repair and maintenance without affecting its operation.
"However, our number one priority is public safety and until we are completely satisfied the bridge is safe to use, it will remain closed.
"The bridge is still under guarantee by the main contractor, who will report back to the council once they have established the cause following their investigations and independent testing of the broken part.
"We would like to apologise for any inconvenience to the travelling public but I think people will understand why we have taken this course of action."
John Colvin, the night manager at the nearby City Inn hotel, said he heard a loud snapping noise when the cable came down.
He said: "To be honest it was a scary sound, it sounded like a bomb going off.
"We felt the vibrations of it right here. We looked out and someone said one of the supports had gone on the bridge and the whole thing was shaking."
The bridge between Finnieston and Pacific Quay opened in 2006.
Mr Colvin added that the cable snapped at about 2330 GMT, when there were no cars on the bridge.
The structure, which spans 140m, is a tied arch design, carrying four traffic lanes. One lane in each direction is reserved for public transport and there are pedestrian and cycle paths.
Running at an angle across the water, it was the first new road bridge over the river to be built since 1969 when it was built at a cost of £20.3m.
The Saltire Society gave the bridge a civil engineering award last year.
^ Back to top | BBC Sport Home | BBC Homepage | Contact us | Help | ©
~RS~q~RS~~RS~z~RS~21~RS~)