There is no "official" ending date for the Civil Rights Movement the way there is for WWII (for example). Many people would argue that the movement continues today because non-whites are still disproportionately poor and suffer from other inequalities.
There are three dates that might be given as the end of the movement, but again, none of these is official.
You could argue that it ended with the passage of the 1964 Civil Rights Act. That law outlawed most forms of segregation and was one of the major goals of the movement.
You could argue that it ended the next year with the passage of the Voting Rights Act of 1965. This was the last major law having to do with rights. (As opposed to later efforts that have centered around economic issues and things like de facto segregation.)
You could argue that it ended in 1968 when its main leader, Martin Luther King, Jr. was assassinated.
Most people would say that the Movement ended sometime in the 1960s, but you could also argue that it continues to this day.