We observed two secondary eclipses of the exoplanet WASP-12b using the Infrared Array Camera on the Spitzer Space Telescope. The close proximity of WASP-12b to its G-type star results in extreme tidal forces capable of inducing apsidal precession with a period as short as a few decades. This precession would be measurable if the orbit had a significant eccentricity, leading to an estimate of the tidal Love number and an assessment of the degree of central concentration in the planetary interior. An initial ground-based secondary-eclipse phase reported by López-Morales et al.(0.510±0.002) implied eccentricity at the 4.5 σ level. The spectroscopic orbit of Hebb et al. has eccentricity 0.049±0.015, a 3σ result, implying an eclipse phase of 0.509±0.007. However, there is a well-documented tendency of spectroscopic data to overestimate small eccentricities. Our eclipse phases are 0.5010±0.0006 (3.6 and 5.8 μm) and 0.5006±0.0007 (4.5 and 8.0 μm). An unlikely orbital precession scenario invoking an alignment of the orbit during the Spitzer observations could have explained this apparent discrepancy, but the final eclipse phase of López-Morales et al.(0.510±+ 0.007− 0.006) is consistent with a circular orbit at better than 2σ. An orbit fit to all the available transit, eclipse, and radial-velocity data indicates precession at< 1σ; a non-precessing solution fits better. We also comment on analysis and reporting for Spitzer exoplanet data in light of recent re-analyses.