  Heavy carbon doping of MgB$_2$ is studied by first principles electronic
structure studies of two types, an ordered supercell
(Mg(B$_{1-x}$C$_{x}$)$_{2}$,x=0.0833) and also the coherent potential
approximation method that incorporates effects of B-C disorder. For the ordered
model, the twofold degenerate $\sigma$-bands that are the basis of the high
temperature superconductivity are split by 60 meV (i.e.7 meV/%C) and the
$\sigma$ Fermi cylinders contain 0.070 holes/cell, compared to 0.11 for
MgB$_2$. A virtual crystal treatment tends to overestimate the rate at which
$\sigma$ holes are filled by substitutional carbon. The coherent potential
approximation (CPA) calculations give the same rate of band filling as the
supercell method. The occupied local density of states of C is almost identical
to that of B in the upper 2 eV of the valence bands, but in the range -8 eV to
-2 eV, C has a considerably larger density of states. The calculations indicate
that the $\sigma$ Fermi surface cylinders pinch off at the zone center only
above the maximum C concentration x ~ 0.10. These results indicate that
Mg(B$_{1-x}$C$_{x}$)$_{2}$ as well as Mg$_{1-x}$Al$_x$B$_2$ is a good system in
which to study the evolution of the unusual electron-phonon coupling character
and strength as the crucial $\sigma$ hole states are filled.
