  We present a detailed study of the quasiparticle contribution to the
low-temperature specific heat of an extreme type-II superconductor at high
magnetic fields. Within a T-matrix approximation for the self-energies in the
mixed state of a homogeneous superconductor, the electronic specific heat is a
linear function of temperature with a linear-$T$ coefficient $\gamma_s(H)$
being a nonlinear function of magnetic field $H$. In the range of magnetic
fields $H\agt (0.15-0.2)H_{c2}$ where our theory is applicable, the calculated
$\gamma_s(H)$ closely resembles the experimental data for the borocarbide
superconductor YNi$_2$B$_2$C.
