We carry out the canonical quantization of the Levi-Civit�� family of static and cylindrical solutions. The reduced phase space of this family of metrics is proved to coincide with that corresponding to the Kasner model, including the associated symplectic structures, except for that the respective domains of definition of one of the phase space variables are not identical. Using this result, we are able to construct a quantum model that describes spacetimes of both the Levi-Civit�� and the Kasner type, and in which the three-dimensional spatial topology is not uniquely fixed. Finally, we quantize to completion the subfamily of Levi-Civit�� solutions which represent the exterior gravitational field of a straight cosmic string. These solutions are conical geometries,ie, Minkowski spacetime minus a wedge. The quantum theory obtained provides us with predictions about the angular size of this wedge.

18 pages, latex, no figures