  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.
