According to this site, the rocket in question was an Vanguard X-248 rocket motor.
I have then dug up an contemporary news article, published in the Monroe News Star on Tuesday, April 14th, 1964 on the front page:
This information (X-248 and that the payload was the OSO spacecraft) led me to the document "HISTORY OF ORBITING SOLAR OBSERVATORY OSO-2", where Section 2 ("The OSO-B Disaster") describes the accident in this way:
2.2 THE DISASTER
Between 0930 and 0939 hours EST on 14 April 1964, the third stage X-248
A-6 solid propellant rocket motor inadvertently ignited and burned in the Spin
Test Facility at Cape Kennedy. The rocket motor with the spacecraft attached
tore loose from the alignment fixture in which it was mounted and shot to the
ceiling of the facility. When it hit the ceiling, the spacecraft was torn loose
from the third stage motor and fell to the floor. The rocket motor continued
on to the corner of the building and burned until its fuel was expended. Eleven
men were burned - three fatally and eight others suffered injuries ranging from
critical to minor. The three men who died were not killed immediately but
died as a result of their burns within a couple days to a couple weeks after the
Eye witness interviews after the accident indicated that the Douglas personnel had just completed their ordnance checks of the third stage/spacecraft combination. One of the Ball Brothers Research employees stepped over to the spacecraft to adjust the polyethelene shroud which was placed over the spacecraft and third stage as a dust protector and to purge them with nitrogen. As he touched the shroud a crackle was heard and the third stage ignited.
Another recap can be found in FLIGHT International, 14 May 1964, Pg. 825 (archived here), where it is called "The X-248 accident".