We formulate statistical mechanics for a mixed diamond chain with spins 1 and 1/2. Owing to a series of conservation laws, any eigenstate of this system is decomposed into eigenstates of finite odd-length spin-1 chains. The ground state undergoes five quantum phase transitions with varying 1, a parameter that controls frustration. We evaluated the residual entropy and Curie constant which characterize each phase and phase boundary at low temperatures. We further find various characteristic finite-temperature properties such as the nonmonotonic temperature dependence of magnetic susceptibility, the multipeak structure in the lambda-dependence of entropy, the plateau-like temperature dependence of entropy and the multipeak structure of specific heat.