We present the first broadband λ = 1 mm spectrum toward the z = 2.56 Cloverleaf quasar, obtained with Z-Spec, a grating spectrograph on the 10.4 m Caltech Submillimeter Observatory. The 190-305 GHz observation band corresponds to the rest frame 272-444 μm, and we measure the dust continuum as well as all four transitions of carbon monoxide (CO) lying in this range. The power-law dust emission, F_ν = 14 mJy(ν/240 GHz)^(3.9) is consistent with the published continuum measurements. The CO J = 6 → 5, J = 8 → 7, and J = 9 → 8 measurements are the first, and now provide the highest-J CO information in this source. Our measured CO intensities are very close to the previously published interferometric measurements of J = 7 → 6, and we use all available transitions and our ^(13)CO upper limits to constrain the physical conditions in the Cloverleaf molecular gas disk. We find a large mass (2-50 × 10^9 M_⊙) of highly excited gas with thermal pressure nT > 10^6 K cm^(–3). The ratio of the total CO cooling to the far-IR dust emission exceeds that in the local dusty galaxies, and we investigate the potential heating sources for this bulk of warm molecular gas. We conclude that both UV photons and X-rays likely contribute, and discuss implications for a top-heavy stellar initial mass function arising in the X-ray-irradiated starburst. Finally, we present tentative identifications of other species in the spectrum, including a possible detection of the H_2O 2_(0,2) → 1_(1,1) transition at λ_(rest) = 303 μm.