  A spectrum generating algebra is constructed and used to find all the
physical states of the $W_3$ string with standard ghost number. These states
are shown to have positive norm and their partition function is found to
involve the Ising model characters corresponding to the weights 0 and 1/16. The
theory is found to be modular invariant if , in addition, one includes states
that correspond to the Ising character of weight 1/2. It is shown that these
additional states are indeed contained in the cohomology of $Q$.
