The Brain from Planet Arous (1957)

The Brain from Planet Aros. The cosmic brain has taken control of John Agar who, in retaliation, buried an axe into the brain’s Fissure of Rolando.

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From Wikipedia:

The Brain from Planet Arous is a 1957 science-fiction film that features the theme of alien possession.

An outer-space terrorist from a planet named Arous – a brain-shaped creature named Gor – arrives on Earth and possesses young scientist Steve March. Gor then proceeds to use his vast, destructive powers to bend the world to his will, threatening to wipe out the capital city of any nation that defies him.

Meanwhile, another brain from Arous – named Vol – arrives on Earth and eventually inhabits the body of March’s fiancee’s dog. Vol explains that Gor is a wanted criminal on their world. Gor’s only weakness is the Fissure of Rolando and he is only vulnerable during one brief period when he needs to exit his host to absorb oxygen.