We present observations of the Sunyaev–Zel’dovich effect (SZE) in the Bullet cluster (1E 0657−56) using the APEX-SZ instrument at 150 GHz with a resolution of 1´. The main results are maps of the SZE in this massive, merging galaxy cluster. The cluster is detected with 23σ significance within the central 1´ radius of the source position. The SZE map has a broadly similar morphology to that in existing X-ray maps of this system, and we find no evidence for significant contamination of the SZE emission by radio or IR sources. In order to make simple quantitative comparisons with cluster gas models derived from X-ray observations, we fit our data to an isothermal elliptical β model, despite the inadequacy of such a model for this complex merging system. With an X-ray-derived prior on the power-law index, β = 1.04⁺⁰·¹⁶ ₋₀.₁₀ , we find a core radius rc = 142´´ ± 18´´, an axial ratio of 0.889 ± 0.072, and a central temperature decrement of −771 ± 71 μKсмв, including a ±5.5% flux calibration uncertainty. Combining the APEX-SZ map with a map of projected electron surface density from Chandra Xray observations, we determine the mass-weighted temperature of the cluster gas to be Tmg = 10.8 ± 0.9 keV, significantly lower than some previously reported X-ray spectroscopic temperatures. Under the assumption of an isothermal cluster gas distribution in hydrostatic equilibrium, we compute the gas mass fraction for prolate and oblate spheroidal geometries and find it to be consistent with previous results from X-ray and weak-lensing observations. This work is the first result from the APEX-SZ experiment, and represents the first reported scientific result from observations with a large array of multiplexed superconducting transition-edge sensor bolometers.