When did the Republicans first become the party of the bigot, and the Democrats become the civil rights party?

We know that in the mid 19th century in the US, the Republicans were in favor of civil rights and the Democrats were against them. At some point, of course, that switched completely. There is no debate that the great civil rights politicians of the 20th century (FDR, JFK, RFK, LBJ, etc) were all Democrats, while their foils (Strom Thurmond, Nixon, Jesse Helms and the like) were all Republican. When and how did this "Freaky Friday"-like transformation take place?