In a statement, SpaceX blamed the blast on “an anomaly” and said no-one had been injured.
The company said the rocket’s payload, a satellite due to be launched on September 3, was also destroyed.
The force of the blast shook buildings several miles away and sent a plume of smoke high above the complex.
SpaceX is seeking to create a new era of reusable rockets and affordable private space travel and has used its Falcon-9 rocket to take supplies to the International Space Station (ISS).
In December 2015, SpaceX successfully landed a Falcon-9 back on Earth after a mission – a first in rocketry.
It went on to recover five more boosters, with most of them touching down on an ocean platform.
The idea is to re-fly these rockets, and the first such “second hand” vehicle is scheduled to launch in October.
SpaceX is run out of Hawthorne near Los Angeles by Elon Musk, who made his fortune with internet companies.
As well as being the rocket company’s CEO, Elon Musk also heads up the Tesla electric car company.
According to Cape Canaveral Air Force Station, a “significant” explosion happened just after 09:00 (14:00 GMT) at Launch Complex 40, which is leased by SpaceX.
Spokesman Bryan Purtell said: “Our emergency management teams are responding right now.”
The Falcon-9 rocket was carrying the Israeli-built Amos-6 communications satellite.
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