The blast went off about 70 yards from a building housing the Chechen police department for fighting organized crime, said Akhmed Dzheirkhanov, the deputy chief of the Emergency Situations Ministry branch for Chechnya. He said a woman was killed.
Earlier, he said another blast occurred nearby, but it was unclear whether it had caused casualties.
TVS television showed footage of a man, cradling an injured arm, running for help on a street strewn with metal fragments reportedly from a large Kamaz truck. Police and troops came streaming into the street, moving quickly toward the smoke-covered scene of the blast.
The explosions came the day before the Chechen State Council, the republic's temporary legislature, was to have met for its first session in a Grozny building that was hastily built to replace the government headquarters destroyed in a December car bombing. That attack killed at least 70 people.
Earlier this month, a female bomber blew up a bus carrying workers from a Russian air base near Chechnya, killing herself and at least 14 people. Two other suicide bombings in a three-day period inside Chechnya last month killed at least 78 people.
The rebel attacks have undercut the Kremlin's efforts to portray the situation in the war-shattered region as stabilizing. The Russian parliament last month approved a partial amnesty in hopes of encouraging rebels to abandon their fight.
Rebels forced Russian troops out of Chechnya after a 1994-96 war, leaving the republic de facto independent. Russian troops rolled back in fall 1999 after rebels raided a neighboring Russian region and after a series of apartment explosions blamed on rebels.