TImis note is acontinuation of [Nl], Wbere we have discusged tbe unknotting number of knots With rspect tia knot diagrams. Wc wilI show that for every minimum-crossing knot-diagram among ah unknotting-number-one two-bridge knot there exist crossings whose exchangeyields tIme trivial knot, ib tbe tbird Tait conjecture is true.