We consider the diffusive properties of Brownian motion in a biased periodic potential. We relate the effective diffusion coefficient to the solution of two coupled time-independent partial differential equations and solve these equations numerically by the matrix-continued-fraction (MCF) method for intermediate values of the temperature and friction coefficient. The weak-noise limit is explored by numerical simulations of the Langevin equations. Here, we identify the regions of parameters for which the diffusion coefficient exponentially grows with inverse temperature. In particular, we demonstrate that there is a finite range of bias forces for which such a growth is observed (region of giant enhancement of diffusion). We also show that at small forces close to the critical range, the diffusion coefficient possesses a pronounced maximum as a function of temperature. All results can be interpreted in the framework of a simple two-state theory incorporating the transition rates between the locked and running solutions.