The Taliban-claimed assault -- the second carried out by the militant group in the Afghan capital in a week -- triggered chaotic scenes as terrified survivors fled the area scattered with body parts and hospitals were overwhelmed by the large number of wounded.
It came as both the insurgents and the Islamic State group have escalated their attacks on Kabul, one of the deadliest places in Afghanistan for civilians.
"The latest toll has reached 63 dead and 151 wounded," Baryalai Hilali, the director of the government media centre, told reporters.
He warned the death toll might rise as some of the wounded brought to hospitals were in a "critical condition".
The blast happened in an area where several high-profile organisations, including the European Union, have offices. Members of the EU delegation in Kabul were in their "safe room" and there were no casualties, an official told AFP.
The suicide bomber passed through at least one checkpoint in the ambulance, saying he was taking a patient to Jamuriate hospital, an interior ministry spokesman told AFP.
"At the second checkpoint he was recognised and blew his explosive-laden car," Nasrat Rahimi said.
Twenty minutes before the blast an AFP reporter saw police checking ambulances several hundred metres from the scene of the explosion, as the drivers and patients stood on the street. Ambulances are rarely checked in the city.
Photos shared on social media purportedly of the blast, which the presidential palace quickly condemned as a "crime against humanity", showed a huge plume of smoke rising into the sky.
Near the blast site civilians walked through debris- covered streets carrying wounded on their backs as others loaded several bodies at a time into ambulances and private cars to take them to medical facilities around the city.
The Italian NGO Emergency said seven dead and 70 wounded had been taken to its hospital, with its coordinator Dejan Panic tweeting that it had been a "massacre".
A man told Ariana TV he had taken his wounded brother to Jamuriate and Emergency hospitals but had been turned away.
"They are asking people with non-life threatening wounds to go to other hospitals," he said.
Aminullah, whose stationery shop is just metres from where the explosion happened, said the force of the explosion shook the foundations of his building.
"The building shook. All our windows broke. The people are in shock in our market," he told AFP.
A man told Tolo News he was passing the area when the explosion happened.
"There were dozens of people who were killed and wounded. There were pools of blood."
The offices of the High Peace Council, charged with negotiating with the Taliban, are also near the blast site.
"It targeted our checkpoint. It was really huge -- all our windows are broken," Hassina Safi, a member of High Peace Council, told AFP.
"So far we don't have any reports if any of our members are wounded or killed."
A security alert issued on Saturday morning had warned that the Islamic State group was planning "to conduct aggressive attacks" on supermarkets, shops and hotels frequented by foreigners.
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