We introduce an exact replica method for the study of critical systems with quenched bond randomness in two dimensions. For the q-state Potts model, we show that a line of renormalization group fixed points interpolates from weak to strong randomness as q-2 grows from small to large values. This theory exhibits a q-independent sector, and allows at the same time for a correlation length exponent which keeps the Ising value and continuously varying magnetization exponent and effective central charge. These findings appear to solve long-standing numerical and theoretical puzzles, and to illustrate the peculiarities which may characterize the conformal field theories of random fixed points.