  We consider one of the simplest time-dependent backgrounds in M-theory,
describing the shrinking away of the M-theory dimension with the other spatial
dimensions static. As the M-theory dimension becomes small, the situation
becomes well-described by string theory in a singular cosmological background
where the string coupling tends to zero but the $\alpha'$-corrections become
large, near the cosmic singularity. We compute these $\alpha'$-corrections,
both for the background and for linearized perturbations, in heterotic string
theory, and show they may be reproduced by a map from eleven-dimensional
M-theory.
