Here is the definitive answer and I can submit a bibliography for a price!:
Oscar de la Renta partnered with Ben and Gerald Shaw to purchase Jane Derby - a Seventh Avenue ready-to-wear firm known for after 5 wear. Derby was preparing to retire after a thirty year career in fashion and sold her business to de la Renta and the Shaws shortly before she died. Oscar de la Renta had a secure and well paid position with Elizabeth Arden, but the opportunity to own a third of Jane Derby and to turn it into his own company was too good an opportunity to miss. Oscar de la Renta had been designing for other people so his name was still unknown and so he used the label 'Oscar de la Renta for Jane Derby' for several years while he made a name for himself. The exact date of when he dropped the 'Jane Derby' from the label is not defined anywhere I could find.
In 1967, de la Renta married Françoise de la Langlade, editor-in-chief of French Vogue. With her help he received important publicity and high end clientelle. Later that year he received a Coty award and in 1968 he received another Coty, as well as a Neiman Marcus award. One reference says he produced his first Oscar de la Renta collection without the use of Jane Derby's name in 1968, however, the name of Jane Derby's company was not officially changed to Oscar de la Renta until 1969 when fashion giant Richton International (aka Coro of costume jewellery fame) bought de la Renta's company. For the next four years de la Renta served on Richton's board of directors and was the CEO of four divisions of the company: Oscar de la Renta, Oscar de la Renta II, Oscar de la Renta Furs, and Oscar de la Renta Jewelry. Oscar de la Renta was the first fashion company in the U.S. to offer its stock to the public, however, in 1973 he reclaimed ownership of his organization and it became Oscar de la Renta Limited.
So it sounds like the business was a bit messy for a few years. I can see why the company skips over the details and just says he was designing under his own name since 1965, because he was, but he did use the 'for Jane Derby' label until 1968 or 1969 when he was better known.