  We investigate the fate of the classical singularity in a collapsing dust
cloud. For this purpose, we quantize the marginally bound Lemaitre-Tolman-Bondi
model for spherically-symmetric dust collapse by considering each dust shell in
the cloud individually, taking the outermost shell as a representative. Because
the dust naturally provides a preferred notion of time, we can construct a
quantum mechanical model for this shell and demand unitary evolution for wave
packets. It turns out that the classical singularity can generically be avoided
provided the quantization ambiguities fulfill some weak conditions. We
demonstrate that the collapse to a singularity is replaced by a bounce followed
by an expansion. We finally construct a quantum corrected spacetime describing
bouncing dust collapse and calculate the time from collapse to expansion.
