We study the existence and uniqueness of nonnegative singular solution u(x,t) of the semilinear parabolic equation $$ u_t = Delta u - a cdot abla(u^q) = u^p, $$ defined in the whole space $R^N$ for t > 0, with initial data $Mdelta(x)$, a Dirac mass, with M > 0. The exponents p,q are larger than 1 and the direction vector a is assumed to be constant. We here show that a unique singular solution exists for every M > 0 if and only if 1 < q < (N + 1)/(N - 1) and 1 < p < 1 + $(2q^*)$/(N + 1), where $q^* = max{q, (N + 1)/N}$. This result agrees with the earlier one for N = 1. In the proof of this result, we also show that a unique singular solution of a diffusion-convection equation without absorption, $$ u_t = Delta u - a cdot abla(u^q), $$ exists if and only if 1 < q < (N + 1)/(N - 1).