We present {em Spitzer} secondary-eclipse observations of the hot Jupiter HAT-P-13 b in the 3.6 {micron} and 4.5 {micron} bands. HAT-P-13 b inhabits a two-planet system with a configuration that enables constraints on the planets second Love number, math{ksb{2}}, from precise eccentricity measurements, which in turn constrains models of the planets interior structure. We exploit the direct measurements of math{e cos omega} from our secondary-eclipse data and combine them with previously published radial velocity data to generate a refined model of the planets orbit and thus an improved estimate on the possible interval for math{ksb{2}}. We report eclipse phases of math{0.49154 pm 0.00080} and math{0.49711 pm 0.00083} and corresponding math{e cos omega} estimates of math{-0.0136 pm 0.0013} and math{-0.0048 pm 0.0013}. Under the assumptions of previous work, our estimate of math{ksb{2}} of 0.81 {pm} 0.10 is consistent with the lower extremes of possible core masses found by previous models, including models with no solid core. This anomalous result challenges both interior models and the dynamical assumptions that enable them, including the essential assumption of apsidal alignment. We also report eclipse depths of 0.081% {pm} 0.008% in the 3.6 {micron} channel and 0.088 % {pm} 0.028 % in the 4.5 {micron} channel. These photometric results are non-uniquely consistent with solar-abundance composition without any thermal inversion.
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