From Wikipedia:
The Last Unicorn is a 1982 American animated fantasy film about a unicorn who, upon learning that she is the last of her species on Earth, goes on a quest to find out what has happened to others of her kind. Based on the 1968 novel The Last Unicorn written by Peter S. Beagle, who also wrote the film’s screenplay, the film was directed and produced by Arthur Rankin Jr. and Jules Bass. It was produced by Rankin/Bass Productions for ITC Entertainment and animated by Topcraft.
The logo is based on Eckmann-Schrift, with an added contour. For the all-caps setting of “Unicorn”, a more conventional form of I was used. The typeface originally came with an alternate form for this character, but this one is different. The movie credits on the poster are set in Helvetica.