The dynamic structure factor f(q,t) of suspensions of adhesive colloidal spheres has been characterized as a function of temperature over the volume fraction range 0.010<φ<0.075. Below a critical temperature that is volume fraction dependent, the suspensions underwent an abrupt, reversible transition in dynamic structure. Below their gel points suspensions became nonergodic, and the time decay of f(q,t) was arrested by as many as five decades. Static light scattering demonstrated that the adhesive spheres formed a fractal cluster gel structure. A recent model of the dynamics of fractal clusters [A. H. Krall and D. A. Weitz, Phys. Rev. Lett. 80, 778 (1998)] was applied to extract the temperature and volume fraction dependence of the characteristic decay times of f(q,t). Immediately above the gel temperature a single stretched exponential decay of f(q,t) was observed. The temperature dependence of the decay time was τα~ɛ-1.15+/-0.06, where ɛ=(T-Tgel)/Tgel, and Tgel is the gelation temperature. The argument of the stretched exponential decay p decreased monotonically as the temperature was lowered toward the gel point, until, at gelation, p~0.5. Below the gel temperature, an initial stretched exponential decay of f(q,t) was followed by a plateau. Finally, at long times, an additional exponential decay of the gel f(q,t) was observed. By applying the fractal cluster dynamics model, it was found that the initial decay time, τβ~ɛ-1.00+/-0.07. The plateau in f(q,t) was due to an upper bound of the mean-squared displacement of gel segments, denoted δ2. The typical magnitude of δ2 was not much greater than the square of the particle radius. The data showed δ2~ɛ-1.05+/-0.07. The additional exponential decay at long times, τγ, depended only weakly on ɛ. Its dependence on the scattering vector was τγ~q-0.53+/-0.06. The argument of the stretched exponential decay of the gel f(q,t) and volume fraction dependence of τβ and δ2 indicate that the spatial scaling of the gel compliance is consistent with the gel network bonds possessing angular rigidity. The ɛ dependence of the characteristic times τα and τβ could not be fully explained by the fractal cluster dynamics model. The long time decay of f(q,t) exhibited behavior that differed from that recently reported for dilute gels of aqueous colloidal polystyrene [Cipelletti et al., Phys. Rev. Lett. 84, 2275 (2000)]. We hypothesize that the long-time decay in f(q,t) of the gels studied here is due to rare bond disaggregation processes that occur because of the relatively weak interaction between the adhesive spheres (ΔEmin/kT~10) of the thermoreversible gel.