  We calculate the spin-drag transresistivity $\rho_{\uparrow \downarrow}(T)$
in a two-dimensional electron gas at temperature $T$ in the random phase
approximation. In the low-temperature regime we show that, at variance with the
three-dimensional low-temperature result [$\rho_{\uparrow\downarrow}(T) \sim
T^2$], the spin transresistivity of a two-dimensional {\it spin unpolarized}
electron gas has the form $\rho_{\uparrow\downarrow}(T) \sim T^2 \ln T$. In the
spin-polarized case the familiar form $\rho_{\uparrow\downarrow}(T) =A T^2$ is
recovered, but the constant of proportionality $A$ diverges logarithmically as
the spin-polarization tends to zero. In the high-temperature regime we obtain
$\rho_{\uparrow \downarrow}(T) = -(\hbar / e^2) (\pi^2 Ry^* /k_B T)$ (where
$Ry^*$ is the effective Rydberg energy) {\it independent} of the density.
Again, this differs from the three-dimensional result, which has a logarithmic
dependence on the density. Two important differences between the spin-drag
transresistivity and the ordinary Coulomb drag transresistivity are pointed
out: (i) The $\ln T$ singularity at low temperature is smaller, in the Coulomb
drag case, by a factor $e^{-4 k_Fd}$ where $k_F$ is the Fermi wave vector and
$d$ is the separation between the layers. (ii) The collective mode contribution
to the spin-drag transresistivity is negligible at all temperatures. Moreover
the spin drag effect is, for comparable parameters, larger than the ordinary
Coulomb drag effect.
