First there was a small explosion, which detonated a bigger bomb, in an abandoned car on Wednesday in a busy street in Srinagar, the summer capital of India's Jammu-Kashmir state, Hari Lal, an officer of the Central Reserve Police Force, said.
The nearly simultaneous blasts left three soldiers and three civilians dead, Lal said. An additional 30 civilians and 10 soldiers were wounded in the explosions, which damaged several vehicles.
Seven people were also killed in violence elsewhere in Kashmir on Wednesday.
An Indian army patrol was attacked by suspected rebels, leaving two soldiers dead in Chunti-Mulla village, 70km north of Srinagar, a police officer said on condition of anonymity.
The attackers allegedly fled after the ambush.
Indian soldiers, meanwhile, intercepted and shot dead three people who were attempting to infiltrate into the Indian-controlled territory from Pakistan's portion of Kashmir early Wednesday near Tangdhar sector, north of Srinagar, according to India's defence ministry spokesman Lt-Col VK Batra.
A school teacher and a former rebel were also gunned down in two incidents, police said.
The Srinagar explosions shattered windows at dozens of nearby buildings, and the twisted metal of the destroyed vehicle was scattered over 50m.
A person identifying himself as the spokesman for a Pakistan-based group, Al-Nasireen, claimed responsibility for the attack in a telephone call to the Current News Service, a local news agency.
Separatists have vowed to continue fighting Indian forces despite India and Pakistan declaring a cease-fire in Kashmir and holding talks aimed at settling the decades-old Kashmir dispute.
Indian forces say they killed
three people on the border
Rebels want to be part of the India-Pakistan dialogue that began in January last year.
Authorities say that nearly 160 fighters, including 38 commanders, have been killed in the Indian-controlled portion of Kashmir since January.
Loss of leaders
"The loss of their leaders has put tremendous pressure on the militants. There is desperation among their ranks, which makes them commit such actions," said K Srinivasan, the intelligence chief of India's paramilitary Border Security Force.
Srinivasan also said that despite peace initiatives by India and Pakistan, at least 50 fighters had entered Indian-held Kashmir from Pakistani territory since January, and another 400 were waiting to cross the Line of Control that divides Kashmir between the neighbouring countries.
Both India and Pakistan claim the Himalayan territory in its entirety.
More than 66,000 people have lost their lives in the conflict.