We obtain the global existence and uniqueness result for a one-dimensional back- ward stochastic Riccati equation, whose generator contains a quadratic term of L (the second unknown component). This solves the one-dimensional case of Bismut- Peng's problem which was initially proposed by Bismut (1978) in the Springer yellow book LNM 649. We use an approximation technique by constructing a sequence of monotone generators and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean-variance hedging problem with stochastic market conditions.