  The null string equations of motion and constraints in the Schwarzschild
spacetime are given. The solutions are those of the null geodesics of General
Relativity appended by a null string constraint in which the "constants of
motion" depend on the world-sheet spatial coordinate. Because of the extended
nature of a string, the physical interpretation of the solutions is completely
different from the point particle case. In particular, a null string is
generally not propagating in a plane through the origin, although each of its
individual points is. Some special solutions are obtained and their physical
interpretation is given. Especially, the solution for a null string with a
constant radial coordinate $r$ moving vertically from the south pole to the
north pole around the photon sphere, is presented. A general discussion of
classical null/tensile strings as compared to massless/massive particles is
given. For instance, tensile circular solutions with a constant radial
coordinate $r$ do not exist at all. The results are discussed in relation to
the previous literature on the subject.
