Ambulances rushed to the scene of the blast, which shattered windows and sent up a huge cloud of smoke visible throughout much of the city, witnesses said.
The Britain-based activist group the Syrian Observatory for Human Rights said at least 42 people were killed, most of them civilians. Some members of the Syrian security forces were also killed, it said.
Syrian state TV called it a "terrorist" attack by a suicide bomber. It said at least 35 people were killed and more than 200 wounded. The state news agency published photos of two dead bodies lying in the street.
There was no way to immediately reconcile the differing death tolls.
Russia's state owned RIA Novosti news agency quoted a Russian Embassy official as saying the Embassy building had been damaged in the blast but no one was hurt.
In a separate attack, Syrian state TV said mortar shells exploded near the Syrian Army General Command in central Damascus, causing no casualties. The station said the building was empty because it was under renovation.
The Britain-based activist group Syrian Observatory for Human Rights said two mortar rounds struck near the building but did not report casualties.
On Wednesday, two mortar shells exploded next to a soccer stadium in Damascus, killing one player. The day before, two mortar shells blew up near one of Assad's three palaces in the city, causing only material damage.
Between the car bomb and the mortar attack near the army command, a security official reported another blast in the capital's northeastern Barzeh neighborhood. He had no other information and spoke on condition of anonymity because he was not authorized to brief the media.
The Observatory, which relies on a network of anti-regime activists inside Syria, said two car bombs had exploded near security centers in Barzeh, followed by intense clashes between rebels and security forces.