The Alfred Hitchcock Hour - Season 1 Episode 23 The Lonely Hours

Louise Henderson rents a room to woman named Vera Brandon who begins to show an unreasonable amount of interest in Louise's newborn son. Louise begines investigating and discovers that she and Vera shared the same maternity ward when her son was born. She also discovers that Vera's baby died during childbirth and this death has left her mentally disturbed. She comes home only to both Vera and her son missing. She finds an address and calls the police. The police go to the address and discover Vera rocking the baby back and forth and claiming that the child is hers. The police return the child to Louise and send Vera to a mental hospital for treatment.