Monte Carlo calculations have been used to study the behavior of a simple cubic spin-\textonehalf{} Ising model with spatially anisotropic nearest-neighbor coupling. The interaction in the
x and
y directions, which is equal to (${J}_{\ensuremath{\perp}}$), was systematically decreased compared to that in the
z direction (${J}_{\ensuremath{\parallel}}$), and results were obtained for values of $\ensuremath{\Delta}=\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}$ as small as 0.003. For small $\ensuremath{\Delta}$ the system behaves like a one-dimensional Ising chain at high temperature before undergoing a low-temperature transition to a three-dimensionally ordered state. We find that the asymptotic ($\ensuremath{\Delta}\ensuremath{\rightarrow}0$) variation of
Tc is accurate over a surprisingly wide range of $\ensuremath{\Delta}$.