The bomber detonated an explosive vest after police tried to stop him entering the Palace of Justice, which is close to the famous Souk al-Hamidieyh market.
Two hours later, a second bomb blast was reported in Rabweh, to the west.
The bombings come on the sixth anniversary of the uprising against President Bashar al-Assad.
What began as an Arab Spring uprising against an autocratic ruler has mushroomed into a brutal proxy war that has drawn in regional and world powers.
Activists say more than 320,000 people have been killed and 11 million displaced.
Police said the suicide bomber blew himself up at 13:20 (11:20 GMT) on Wednesday, as officers tried to search him at the entrance to the Palace of Justice.
Ahmed al-Sayyid, a senior legal official, told state TV that the area was crowded with lawyers, judges and civilians at the time.
"We were terrified because the sound of the explosion was enormous," a lawyer who was inside the complex during the attack told AFP news agency.
"We took refuge in the library which is on a higher floor," the lawyer added, speaking on condition of anonymity. "It was a bloody scene."
An AFP correspondent reported that the area has been locked down and that roads leading to it were blocked by ambulances and fire engines.
There were no immediate reports of any casualties from the second bombing in Rabweh - an area about 2km from the scene of the first attack.
No group has said yet that it carried out the bombings, the latest in a series that have targeted government-controlled areas.
Source : BBC
