The Slender Thread(1965)

Overview:

Alan is a Seattle college student volunteering at a crisis center. During a lonely night shift, a woman calls up and tells Alan that she needs to talk to someone. She informs him she's taken an overdose of pills; he tries desperately to get help to her, while keeping her on the line. As crucial minutes tick by, Alan learns more about the woman, her family life, and why she wants to die.