The Trapezium cluster in the Orion Nebula has been monitored on 25 nights from November of 1990 to March of 1991, with a CCD attached to the Perkin telescope at Van Vleck Observatory. Photometry in the I band of more than 150 stars has been obtained, and 28 showed significant variation during our study. A periodogram analysis reveals that seven of these have significant peaks in their power spectra, which are interpreted as the rotation periods of the stars. Periods range from 6.2 to 14.3 days. Three of them have periods longer than any previously detected for pre-main-sequence objects, and the group average, 8.7 days, is more than twice as long as the average period of previously measured T Tauri stars. Three possible explanations for this are considered: (1) There is a real difference in the rotational velocities of Trapezium cluster stars and T Tauri stars in associations, perhaps because the Trapezium cluster stars are extremely young; (2) selection effects are hindering the discovery of longer rotation periods among T Tauri stars in associations, and (3) selection effects are hindering the discovery of short-period stars in the Trapezium cluster. Classical and weak T Tauri stars clearly coexist in this extremely young cluster, but the classical stars appear to be confined to I magnitudes fainter than 12.5.