We present a new formulation of the tensionless string (T=0) where the
space-time conformal symmetry is manifest. Using a Hamiltonian BRST scheme we
quantize this {\em Conformal String} and find that it has critical dimension
D=2. This is in keeping with our classical result that the model describes
massless particles in this dimension. It is also consistent with our previous
results which indicate that quantized conformally symmetric tensionless strings
describe a topological phase away {}from D=2. We reach our result by
demanding nilpotency of the BRST charge and consistency with the Jacobi
identities. The derivation is presented in two different ways: in operator
language and using mode expansions. Careful attention is payed to
regularization, a crucial ingredient in our calculations.Comment: 33pp (LaTeX), USITP-94-0