As Draper signs his contract (with the date 7/23/1963, the number referenced in the episode title), the last part of the company name can be seen in the signatory section, and it ends with "L.L.C." Limited Liability Companies weren't in common use in the US until the late 1970s, and prior to that many states had no laws governing their existence.
The first edition of the General Mills Hostess Cookbook that is shown on the counter top in Betty's kitchen wasn't released until 1967, this point in time is 1963 as shown when Don signs his contract in front of Burt Cooper at the end of the show.
Much is made of the fact that Don Draper does not have a "contract". However, early in the series (at least two years before he meets Conrad Hilton, who forces the issue) Don is made a partner with 12.5% of the firm.
The paper work surrounding this share offering/grant is a de-facto contract, regardless of anything else associated with Don's work