© 2015 American Physical Society. The cosmic microwave background (CMB) energy spectrum is a near-perfect blackbody. The standard model of cosmology predicts small spectral distortions to this form, but no such distortion of the sky-averaged CMB spectrum has yet been measured. We calculate the largest expected distortion, which arises from the inverse Compton scattering of CMB photons off hot, free electrons, known as the thermal Sunyaev-Zel'dovich (TSZ) effect. We show that the predicted signal is roughly one order of magnitude below the current bound from the COBE-FIRAS experiment, but it can be detected at enormous significance (1000σ) by the proposed Primordial Inflation Explorer (PIXIE). Although cosmic variance reduces the effective signal-to-noise ratio to 230σ, this measurement will still yield a subpercent constraint on the total thermal energy of electrons in the observable Universe. Furthermore, we show that PIXIE can detect subtle relativistic effects in the sky-averaged TSZ signal at 30σ, which directly probe moments of the optical depth-weighted intracluster medium electron temperature distribution. These effects break the degeneracy between the electron density and the temperature in the mean TSZ signal, allowing a direct inference of the mean baryon density at low redshift. Future spectral distortion probes will thus determine the global thermodynamic properties of ionized gas in the Universe with unprecedented precision. These measurements will impose a fundamental "integral constraint" on models of galaxy formation and the injection of feedback energy over cosmic time.