The first analysis of a transverse loop oscillation observed by both Solar TErrestrial RElations Observatories (STEREO) spacecraft is presented, for an event on the 2007 June 27 as seen by the Extreme Ultraviolet Imager (EUVI). The three-dimensional loop geometry is determined using a three-dimensional reconstruction with a semicircular loop model, which allows for an accurate measurement of the loop length. The plane of wave polarization is found from comparison with a simulated loop model and shows that the oscillation is a fundamental horizontally polarized fast magnetoacoustic kink mode. The oscillation is characterized using an automated method and the results from both spacecraft are found to match closely. The oscillation period is 630 +/- 30 s and the damping time is 1000 +/- 300 s. Also, clear intensity variations associated with the transverse loop oscillations are reported for the first time. They are shown to be caused by the effect of line-of-sight integration. The Alfven speed and coronal magnetic field derived using coronal seismology are discussed. This study shows that EUVI/STEREO observations achieve an adequate accuracy for studying long-period, large-amplitude transverse loop oscillations.