We present results of high-resolution thermal-expansion and specific-heat measurements on single crystalline alpha'-NaV2O5. We find clear evidence for two almost degenerate phase transitions associated with the formation of the dimerized state around 33 K: A sharp first-order transition at T-1 = (33 +/- 0.1) K slightly below the onset of a second-order transition at T(2)(onset)approximate to(34+/-0.1) K. The latter is accompanied by pronounced spontaneous strains. Our results are consistent with a structural transformation at T-1 induced by the incipient spin-Peierls order parameter above T-2 = T-SP.