We present a measurement of the one-point probability distribution function (PDF) of the thermal Sunyaev-Zel'dovich (tSZ) decrement in the pixel temperature histogram of filtered 148 GHz sky maps from the Atacama Cosmology Telescope (ACT). The PDF includes the signal from all galaxy clusters in the map, including objects below the signal-to-noise threshold for individual detection, making it a particularly sensitive probe of the amplitude of matter density perturbations, $\sigma_8$. We use a combination of analytic halo model calculations and numerical simulations to compute the theoretical tSZ PDF and its covariance matrix, accounting for all noise sources and including relativistic corrections. From the measured ACT 148 GHz PDF alone, we find $\sigma_8 = 0.793 \pm 0.018$, with additional systematic errors of $\pm 0.017$ due to uncertainty in intracluster medium gas physics and $\pm 0.006$ due to uncertainty in infrared point source contamination. Using effectively the same data set, the statistical error here is a factor of two lower than that found in ACT's previous $\sigma_8$ determination based solely on the skewness of the tSZ signal. In future temperature maps with higher sensitivity, the tSZ PDF will break the degeneracy between intracluster medium gas physics and cosmological parameters.