We show that the measured intrinsic octupole moments of ^{220}Rn, ^{224}Ra, and ^{226}Ra constrain the intrinsic Schiff moments of ^{225}Ra, ^{221}Rn, ^{223}Rn, ^{223}Fr, ^{225}Ra, and ^{229}Pa. The result is a dramatically reduced uncertainty in intrinsic Schiff moments. Direct measurements of octupole moments in odd nuclei will reduce the uncertainty even more. The only significant source of nuclear-physics error in the laboratory Schiff moments will then be the intrinsic matrix elements of the time-reversal noninvariant interaction produced by CP-violating fundamental physics. Those matrix elements are also correlated with octupole moments, but with a larger systematic uncertainty.