  {}From the ordinary tensile string we derive a geometric action for the
tensionless ($T=0$) string and discuss its symmetries and field equations. The
Weyl symmetry of the usual string is shown to be replaced by a global
space-time conformal symmetry in the $T\to 0$ limit. We present the explicit
expressions for the generators of this group in the light-cone gauge. Using
these, we quantize the theory in an operator form and require the conformal
symmetry to remain a symmetry of the quantum theory. Modulo details concerning
zero-modes that are discussed in the paper, this leads to the stringent
restriction that the physical states should be singlets under space-time
diffeomorphisms, hinting at a topological theory. We present the details of the
calculation that leads to this conclusion.
