  We report high-resolution, bulk Compton scattering measurements unveiling the
Fermi surface of an optimally-doped iron-arsenide superconductor,
Ba(Fe$_{0.93}$Co$_{0.07}$)$_2$As$_2$. Our measurements are in agreement with
first-principles calculations of the electronic structure, revealing both the
$X$-centered electron pockets and the $\Gamma$-centered hole pockets. Moreover,
our data are consistent with the strong three-dimensionality of one of these
sheets that has been predicted by electronic structure calculations at the
local-density-approximation-minimum As position. Complementary calculations of
the noninteracting susceptibility, $\chi_0({\bf q}, \omega)$, suggest that the
broad peak that develops due to interband Fermi-surface nesting, and which has
motivated several theories of superconductivity in this class of material,
survives the measured three dimensionality of the Fermi surface in this family.
