  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 planet's
second Love number, \math{k\sb{2}}, from precise eccentricity measurements,
which in turn constrains models of the planet's 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 planet's orbit and thus an improved estimate on the
possible interval for \math{k\sb{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{k\sb{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.
