We study the 3D Ising model in the infinite volume limit
$N_{x,y,z}\to\infty$ by means of numerical simulations. We determine $T_c$
as well as the critical exponents $\beta,\gamma$ and $\nu$, based
on finite-size scaling and histogram reweighting techniques. In addition,
we study a ''dimensionally reduced'' scenario where $N_z$ is kept fixed
(e.g. at 2, 4, 8), while the limit $N_{x,y}\to\infty$ is taken. For each
fixed $N_z$ we determine $T_c$ as well as $\beta,\gamma,\nu$. For
$T_c$ we find a smooth transition curve which connects the well known
critical temperatures of the 2D and the 3D Ising model. Regarding
$\beta,\gamma,\nu$ our data suggest that the ''dimensionally
reduced'' Ising model is in the same universality class as the 2D Ising
model, regardless of $N_z$.