  The in-plane thermal conductivity $\kappa$ of overdoped FeAs-based
superconductor BaFe$_{1.73}$Co$_{0.27}$As$_2$ ($T_c$ = 8.1 K) single crystal
was measured down to 80 mK. In zero field, the residual linear term
$\kappa_0/T$ is negligible, suggesting a nodeless superconducting gap in the
$ab$-plane. In magnetic field, $\kappa_0/T$ increases rapidly, very different
from that of conventional s-wave superconductors. This anomalous
$\kappa_0/T(H)$ may reveal an exotic superconducting gap structure in overdoped
BaFe$_{1.73}$Co$_{0.27}$As$_2$: the vanishing hole ($\beta$) pocket has a much
larger gap than the electron ($\gamma$ and $\delta$) pockets which contain most
of the carriers. Such an exotic gap structure is an evidence for
superconducting state induced by interband interactions, in which the band with
the {\it smaller} density of states has a {\it larger} gap.
