Clearing up some of the misstatements and the reposting of errors made on other sites:
The car was conceived by Andrew Di Dia (not Bobby Darin), a clothing designer, who Darin had met while on tour in Detroit in 1957. Di Dia enlisted the help of his neighbor, Chrysler designer Edward V. Francoise, to turn his sketches into a proper design and a clay model of the car that was eventually built. For seven years, from 1953 to 1960 the Di Dia 150 was hand-built by four workers, at a cost of $93,647.29; it was sold to Darin in 1961.
Francoise' sketches were presented to Detroit custom shop Clarkaiser Customs (along with a clay scale model). Unlike the firm’s normal custom jobs, Di Dia’s car was not based on an existing vehicle, it was created from the ground up, using a framework of round chrome-molybdenum tubing fabricated by Clarkaiser’s principals, Ronald Clark and Robert Kaiser. (Not Barris)
When completed, Di Dia towed it behind his Buick to Los Angeles in time for an official debut at George Barris' third annual Bakersfield Motor and Boat Show which was to be held November 24-27, 1960 at the Kern County Fairgrounds (which is the only connection to Barris)
A ’58 Cadillac 365-cubic-inch V8 was fitted into the car using a marine induction manifold that gave the Cadillac a low engine profile. A matching ’58 Cadillac automatic transmission sent power to the rear wheels. It was later replaced by a 427 Ford engine when it was taken on the show circuit
As an added bonus, here is the car with the flip-down headlights showing.