We investigate one-dimensional lattice systems with (symmetric) nearest neighbor transfer ratesWn, n+1 which are independently distributed according to a probability densityρ(w). For two general classes ofρ(w), we rigorously determine the asymptotic behavior of the relevant single site Green function 〈\(\tilde P\)0(ω)〉 nearω=0, and obtain exact results for the long time decay of the initial probability amplitude and for the low energy density of states. A scaling hypothesis, accurately confirmed by computer simulations, is used to relate the low frequency hopping conductivityσ(ω) uniquely to 〈\(\tilde P\)0(−iω)〉, and we conjecture that the resulting asymptotic behavior forσ(ω) is also exact. The critical exponents associated with the various asymptotic laws depend onρ(w) and show a crossover from universal to non-universal behavior. Comparison is made with the results of several approximate treatments.