the moments of the waiting time, we may expect therefore that at nonzero temperatures and f < g$_{0}$ the exponent α and so the character of anomalous diffusion becomes depending on f (the diffusion behavior at f > g$_{0}$ is expected to be normal). Specifically, with decreasing of f from g$_{0}$ to f$_{tr}$ the exponent α should also decrease from 2 to 1, and if f < f$_{tr}$ then α < 1. Of course, in order to find the dependence of α on f and the diffusion laws a quantitative consideration of the problem is needed. It is especially important because the effects arising from the joint action of quenched disorder and thermal fluctuations are often unexpected and even counterintuitive. We have studied in the long-time limit the unidirectional transport of particles which occurs under a constant force in a randomly layered medium. The influence of the layers is modeled by a piecewise constant random force whose values in different layers are assumed to be independent and identically distributed with bounded support. We have reduced the problem of the unidirectional transport, initially formulated in the framework of the motion equation, to a continuous-time random walk on a semi-infinite chain. The main statistical characteristic of this approach, the waiting time probability density, is expressed through the probability density of the random force and particle characteristics, including the particle mass. By analyzing the dependence of the moments of the waiting time on the external force, we have formulated the conditions under which the biased diffusion exhibits the anomalous behavior. It has been shown that this behavior may occur only if the external force is equal to the boundary value of the random force. In order to find in the anomalous regime the long-time behavior of the first and second moments of the particle position, we have used the Tauberian theorem and its modified version allowing, in most cases, to determine the first two terms of the asymptotic expansion of these moments. Within this approach, we have found, with one exception, the explicit asymptotic formulas for the variance of the particle position, i.e., the laws of diffusion. The time dependence of the variance is completely controlled by the exponent describing the asymptotic behavior of the probability density of the random force in the vicinity of its minimum value. It has also been shown that, depending on the value of this exponent, the particle mass in the weakly underdamped regime can either enhance or suppress the V. CONCLUSIONS 17