Churchill Square has been evacuated.
Four fire engines are now at the scene, two from Preston Circus and two from Hove, along with the Incident Command Unit from Lewes.
An acetylene cylinder had caught alight in a storage area where some building work was being carried out.
The fire is now out and crews are cooling the acetylene cylinder using one hose reel jet. The cooling process is expected to take some time.
Churchill Square Shopping Centre will be closed for at least four hours.
More top news
An inquest is due to open into the deaths of five friends who drowned during a day trip to Camber Sands in East Sussex last summer.
Philippa has Monday's weather for the east
Philippa has Monday's weather for the west