  We study a modified bosonic string theory that has a pressureless ``dust''
field on the string worldsheet. The dust is a real scalar field with unit
gradient which breaks conformal invariance. Hamiltonian analysis reveals a time
reparametrization constraint linear in the dust field momentum and a spatial
diffeomorphism constraint. This feature provides a natural ``dust time" gauge
in analogy with the parametrized particle. In this gauge we give a Fock
quantization of the theory, which is complete and self-consistent in d<26. The
Hamiltonian of the theory is not a constraint; as a consequence the Hilbert
space and mass spectrum are characterized by an additional parameter, and
includes the usual string spectrum as a special case. The other sectors provide
new particle spectra, some of which do not have tachyons.
