Aims. Previous secondary eclipse observations of the hot Jupiter Qatar-1b in the Ks band suggest that it may have an unusually high day side temperature, indicative of minimal heat redistribution. There have also been indications that the orbit may be slightly eccentric, possibly forced by another planet in the system. We investigate the day side temperature and orbital eccentricity using secondary eclipse observations with Spitzer. Methods. We observed the secondary eclipse with Spitzer/IRAC in subarray mode, in both 3.6 and 4.5 μm wavelengths. We used pixel-level decorrelation to correct for Spitzer’s intra-pixel sensitivity variations and thereby obtain accurate eclipse depths and central phases. Results. Our 3.6 μm eclipse depth is 0.149 ± 0.051% and the 4.5 μm depth is 0.273 ± 0.049%. Fitting a blackbody planet to our data and two recent Ks band eclipse depths indicates a brightness temperature of 1506 ± 71 K. Comparison to model atmospheres for the planet indicates that its degree of longitudinal heat redistribution is intermediate between fully uniform and day-side only. The day side temperature of the planet is unlikely to be as high (1885 K) as indicated by the ground-based eclipses in the Ks band, unless the planet’s emergent spectrum deviates strongly from model atmosphere predictions. The average central phase for our Spitzer eclipses is 0.4984 ± 0.0017, yielding e cos ω = −0.0028 ± 0.0027. Our results are consistent with a circular orbit, and we constrain e cos ω much more strongly than has been possible with previous observations.