In order to characterize more precisely the superconducting transition of a quasi-2D superconductor, we quenched polycrystalline samples of Bi2Sr2CaCu2Ox from various locations of the (T; p(O2)) equilibrium diagram. A well-defined specific heat peak at Tc is observed with 0.01% resolution for the samples with the highest Tc. The samples with lower Tc's provide strong support to the normal-state background determined by an analysis of the data including phonon and fluctuation terms. We do not find any indication of a mean-field jump within experimental uncertainty, in stark contrast to YBa2Cu3O7. The application of a 14 Tesla field depresses the total specific heat by 0.54% only at T(peak)=92.5K, but this effect reduces to less than 0.03% already ±8K away from Tc. On passing the irreversibility line, no anomaly in the specific heat is detected at the 0.01% level.