We have measured the molecular weight distribution (MWD) in a case of equilibrium polymerization. We have studied the time development of the MWD of "living" bifunctional poly(alpha-methylstyrene) in tetrahydrofuran after a quench to 21 K below the polymerization temperature, T-p. We see an intermediate Gaussian distribution evolving toward a final exponential distribution, as expected from theoretical considerations. We see a longer equilibration time for the number average molecular weight (M-n) as well as for the weight average molecular weight (M-w) than for the monomer concentration ([M)], whereas theories predict that M-n and [M] will relax together and that M-w will take much longer. We attribute the delayed equilibration and a second peak at about M-n/4 to the effects of ionic aggregation of the living polymers. We have also studied the equilibrium MWD of this system as a function of the temperature below T-p, and thus as a function of the number average degree of polymerization (L). These measurements and the time study discussed above are the first experimental evidence that the equilibrium MWD for an organic polymer in a state of equilibrium polymerization is an exponential/Flory-Schulz distribution, and is consistent with scaling predictions. Near T-p and at low L, we observe a deviation from the exponential distribution, which may be evidence of the effect of a chain-length dependence of the equilibrium constant for polymerization, or of the effects of polydispersity on correlations due to excluded volume. In addition, the measured L is about two times less than that expected from the initiator concentration; this could result from ionic aggregation or from chain transfer reactions.