June 4 - President John F. Kennedy signs Executive Order 11110, authorizing the Secretary of the Treasury to issue silver certificates.

June 5 - The first annual National Hockey League Entry Draft is held in Montreal, Quebec.

June 10 - President John F. Kennedy delivers his American University speech, "A Strategy of Peace", in Washington, D.C.

June 10 - The University of Central Florida is established by the Florida legislature.

June 11 - In Saigon, Buddhist monk Thich Quang Duc commits self-immolation to protest the oppression of Buddhists by the Ngo Dinh Diem administration. - Alabama Governor George C. Wallace stands in the door of the University of Alabama to protest against integration, before stepping aside and allowing African Americans James Hood and Vivian Malone to enroll. - President John F. Kennedy broadcasts a historic Civil Rights Address, in which he promises a Civil Rights Bill, and asks for "the kind of equality of treatment that we would want for ourselves."

June 12 - Medgar Evers is murdered in Jackson, Mississippi (his killer is convicted in 1994). - Release of the film Cleopatra.

June 13 - The cancellation of Mercury-Atlas 10 effectively ends the United States' manned spaceflight Project Mercury. - The New York Commodity Exchange begins trading silver futures contracts.

June 16 - Vostok 6 carries Soviet cosmonaut Valentina Tereshkova, the first woman into space.

June 17 - Abington School District v. Schempp: The U.S. Supreme Court rules that state-mandated Bible reading in public schools is unconstitutional.

June 19 - Valentina Tereshkova the first woman in space, returns to Earth.

June 20 - Establishment of the Moscow-Washington hotline (officially, the Direct Communications Link or DCL; unofficially, the "red telephone"; and in fact a teleprinter link) is authorized by signing of a Memorandum of Understanding in Geneva by representatives of the Soviet Union and the United States. - Swedish Air Force Colonel Stig Wennerström arrested as a spy for the Soviet Union.

June 21 - Pope Paul VI (Giovanni Battista Montini) succeeds Pope John XXIII as the 262nd pope.

June 26 - John F. Kennedy gives his "Ich bin ein Berliner" speech in West Berlin, East Germany. (source: Wikipedia)