  The application of the conventional saddle-point approximation to condensed
Bose gases is thwarted by the approach of the saddle-point to the ground-state
singularity of the grand canonical partition function. We develop and test a
variant of the saddle-point method which takes proper care of this
complication, and provides accurate, flexible, and computationally efficient
access to both canonical and microcanonical statistics. Remarkably, the error
committed when naively employing the conventional approximation in the
condensate regime turns out to be universal, that is, independent of the
system's single-particle spectrum. The new scheme is able to cover all
temperatures, including the critical temperature interval that marks the onset
of Bose--Einstein condensation, and reveals in analytical detail how this onset
leads to sharp features in gases with a fixed number of particles. In
particular, within the canonical ensemble the crossover from the
high-temperature asymptotics to the condensate regime occurs in an
error-function-like manner; this error function reduces to a step function when
the particle number becomes large. Our saddle-point formulas for occupation
numbers and their fluctuations, verified by numerical calculations, clearly
bring out the special role played by the ground state.
