We have obtained high-quality Keck optical spectra at three epochs of the Type la supernova 1997ex, whose redshift z is 0.361. The elapsed calendar time between the first two spectra was 24.88 days, and that between the first and third spectra was 30.95 days. In an expanding universe where 1 + z represents the factor by which space has expanded between the emission and detection of light, the amount of aging in the supernova rest frame should be a factor of 1/(1 + z) smaller than the observed-frame aging; thus, we expect SN 1997ex to have aged 18.28 and 22.74 days between the first epoch and the second and third epochs, respectively. The quantitative method for determining the spectral-feature age of an SN la reveals that the corresponding elapsed times in the supernova rest frame were 16.97 ± 2.75 and 18.01 ± 3.14 days, respectively. This result is inconsistent with no time dilation with a significance level of 99.0%, providing evidence against "tired light" and other hypotheses in which no time dilation is expected. Moreover, the observed timescale of spectral evolution is inconsistent with that expected in the "variable mass theory." The result is within ∼1 σ of the aging expected from a universe in which redshift is produced by cosmic expansion. © 2005. The American Astronomical Society. All rights reserved.