Godzilla Vs. The Thing (1964)

Starring Akira Takarada, Yuriko Hoshi, Hiroshi Koizumi, Yu Fujiki, Kenji Sahara
Directed by Inoshiro Honda
(actor & director credits courtesy IMDB.com)

A giant storm transports an enormous egg to a Japanese city, which is claimed by greedy industrialists, but its contents are the progeny of Mothra, who will soon face off with Godzilla. 

Although the American title of this film leads you to believe there's a mystery behind the egg, its Japanese title was Mothra Vs. Godzilla and it becomes pretty clear that the egg belongs to Mothra early on, as soon as Emi and Yumi Ito reprise their roles as the tiny princesses of Mothra's island from the creature's first film.  Those hoping for a full-on battle between Godzilla and the flying moth monster may be disappointed as Mothra is weakened and dying during the battle, but there will be others who will carry on.  The filmmakers stage a clever strategic confrontation by Godzilla's foes with some pretty good effects, and the human cast are entertaining, with Takarada returning for his second Godzilla film, although playing a different character.  I haven't yet seen enough Godzilla films to estimate where this ranks in the series, although for me it's clearly behind the original, but still a fun entry, and one of the first to pair off Toho's different kaiju.