We present Herschel Spectral and Photometric Imaging Receiver (SPIRE) Fourier Transform Spectrometer (FTS) observations of the Antennae (NGC 4038/39), a well-studied, nearby (22 Mpc), ongoing merger between two gas-rich spiral galaxies. The SPIRE-FTS is a low spatial (FWHM similar to 19"-43") and spectral (1.2 GHz) resolution mapping spectrometer covering a large spectral range (194-671 mu m, 450-1545 GHz). We detect five CO transitions (J = 4-3 to J = 8-7), both [CI] transitions, and the [N I] 205 mu m transition across the entire system, which we supplement with ground-based observations of the CO J = 1-0, J = 2-1, and J = 3-2 transitions and Herschel Photodetecting Array Camera and Spectrometer (PACS) observations of [C II] and [O I] 63 Am. Using the CO and [CI] transitions, we perform both a local thermodynamic equilibrium (LTE) analysis of [CI] and a non-LTE radiative transfer analysis of CO and [C I] using the radiative transfer code RADEX along with a Bayesian likelihood analysis. We find that there are two components to the molecular gas: a cold (T-kin similar to 10-30 K) and a warm (T-kin greater than or similar to 100 K) component. By comparing the warm gas mass to previously observed values, we determine a CO abundance in the warm gas of x(CO) similar to 5 x 10(-5). If the CO abundance is the same in the warm and cold gas phases, this abundance corresponds to a CO J = 1-0 luminosity-to-mass conversion factor of alpha(co) similar to 7 M circle dot pc(-2) (K km s(-1))(-1) in the cold component, similar to the value for normal spiral galaxies. We estimate the cooling from H-2, [C II], CO, and [O 1] 63 Am to be similar to 0.01 L circle dot/M circle dot. We compare photon-dominated region models to the ratio of the flux of various CO transitions, along with the ratio of the CO flux to the far-infrared flux in NGC 4038, NGC 4039, and the overlap region. We find that the densities recovered from our non-LTE analysis are consistent with a background far-ultraviolet field of strength G(0) similar to 1000. Finally, we find that a combination of turbulent heating, due to the ongoing merger, and supernova and stellar winds are sufficient to heat the molecular gas.