Who was the worlds first ever female Prime Minister?

It was Siramavo Bandaranaike of Ceylon (now Sri Lanka), who died in 2000.

Sirimavo Bandaranaike (1916-2000) was Prime minister of Sri Lanka 3 times:

21 Jul 1960 - 27 Mar 1965

29 May 1970 - 23 Jul 1977

14 Nov 1994 - 10 Aug 2000

She was the first woman to head of government in world history and probably the oldest female political leader in active by the time of her demise in 2000. Her husband, Solomon Bandaranaike, were prime minister in 1956-1959 when he was assassinated, in 1994 she got her third government mandate from her daughter, Chandrika Kumaratunga, who was just sworn in as president of the Republic. That was the first time in history a woman succeeded another woman at the helm of a top State office through elections.