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 ∼ 19″–43″) and spectral (∼1.2 GHz) resolution mapping spectrometer covering a large spectral range (194–671 μm, 450–1545 GHz). We detect five CO transitions (J = 4–3 to J = 8–7), both [C i] transitions, and the [N ii] 205 μm 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 μm. Using the CO and [C i] transitions, we perform both a local thermodynamic equilibrium (LTE) analysis of [C i] 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 (Tkin ∼ 10–30 K) and a warm (Tkin ≳ 100 K) component. By comparing the warm gas mass to previously observed values, we determine a CO abundance in the warm gas of xCO ∼ 5 × 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 αCO ∼ 7 M☉ pc−2 (K km s−1)−1 in the cold component, similar to the value for normal spiral galaxies. We estimate the cooling from H2, [C ii], CO, and [O i] 63 μm to be ∼0.01 L☉/M☉. 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 G0 ∼ 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.