We present a new method called the More-Sorensen Sequential (MSS) method for computing the minimizer of a quadratic function defined by a limited-memory BFGS matrix subject to a two-norm trust-region constraint. This solver is an adaptation of the More-Sorensen direct method into a L-BFGS setting for large-scale optimization. The MSS method uses a recently proposed fast direct method for solving large shifted BFGS systems of equations.