  We present a family of classical spacetimes in 2+1 dimensions. Such a
spacetime is produced by a Nambu-Goto self-gravitating string. Due to the
special properties of three-dimensional gravity, the metric is completely
described as a Minkowski space with two identified worldsheets. In the flat
limit, the standard string is recovered. The formalism is developed for an open
string with massive endpoints, but applies to other boundary conditions as
well. We consider another limit, where the string tension vanishes in
geometrical units but the end-masses produce finite deficit angles. In this
limit, our open string reduces to the free-masses solution of Gott, which
possesses closed timelike curves when the relative motion of the two masses is
sufficiently rapid. We discuss the possible causal structures of our spacetimes
in other regimes. It is shown that the induced worldsheet Liouville mode obeys
({\it classically}) a differential equation, similar to the Liouville equation
and reducing to it in the flat limit. A quadratic action formulation of this
system is presented. The possibility and significance of quantizing the
self-gravitating string, is discussed.
