I just play F, C, Em, and Am throughout the entire song.

F (133211@1)                                   CYou (x32010@1)had my heart and we'll never be world's apart...
Em (022000@1)                                 AmMaybe (x02210@1)in magazines, but you'll still be my star.

Same chords...
             F (133211@1)                       C'Cause (x32010@1)I'm singing in the rain, just singing in the rain
           Em (022000@1)               AmWhat (x02210@1)a glorious feelin', I'm happy again. 
  F (133211@1)                      CI'm (x32010@1)laughing at clouds so dark up above.
     Em (022000@1)                   Am (x02210@1)I'm singing-singing in the rain.