Full size cars such as Impala and Caprice, along with pickup trucks and Corvettes have used frames since day one. It's the econoboxes and mid size cars that started life as unitized construction.
A google search turned up this answer concerning the early use of unibody construction....copied from www.wikipedia.com
" The first automotive application of the monocoque technique was 1923's Lancia Lambda. Chrysler and Citroën built the second monocoque vehicles, both in 1934, with the innovative Chrysler Airflow and the Traction Avant, respectively, and General Motors followed with the Opel Olympia in 1935. In 1936, Lincoln introduced the Zephyr, a monocoque design which was as strong as the Airflow yet much lighter.
Nash Motors introduced this type of construction in 1941 with the new 600, generally credited with being the first popular mass-produced unibody construction automobile made in the United States. The all-welded steel with sturdy bridge-like girders that arched front to rear made for greater strength, safety, and longer life. Nash engineers claimed that about 500 pounds of excess weight was cut out (compared to body-on-frame automobiles) and the body's lower air drag helped it to achieve excellent gas mileage for its day. Prophetically, the company's 1942 news release text attached to the X-ray drawing describes how "... all auto bodies will be built ... as this some day..." The Alec Issigonis Morris Minor of 1948 featured a monocoque body. The Hudson Hornet, along with the rest of Hudson range, featured a monocoque body at the same time.
In the post-war period the technique became more widely used. Other automakers incorporated this type of construction and the terms unit body and unibody became more common in general use. The Ford Consul was the first Ford built in England using a unibody.
In 1960, a major breakthrough in unibody construction was reached in mass-produced Detroit vehicles with the Chevrolet Corvair, the most successful automobile of this type up to that time, with 1,786,243 cars being produced between 1960 and 1969[1][2]. Among its many other forward-thinking and breakthrough technologies for its day, the Corvair was built from uniform molds and relied on the shaping of the glass and doors for help with structural integrity[3]. Convertible versions needed special supports welded underneath to compensate for the missing shape on the top.
American Motors (AMC) continued its engineering heritage from Nash and Hudson with breakthroughs such as in 1963 of combining separate parts into single stampings. The Rambler Classic had "uniside" door surrounds from a single stamping of steel that reduced weight and assembly costs, as well as increasing structural rigidity and improving door fitment."