We consider the problem of minimizing a quadratic function subject to an ellipsoidal constraint when the matrix involved is the Hessian of a quadratic penalty function (i.e., a function of the form $p(x) = f(x) + \frac{1}{2\mu} c(x)^{T} c(x))$. Most applications of penalty functions require $p(x)$ to be minimized for values of $\mu$ decreasing to zero. In general, as $\mu$ tends to zero the nature of finite precision arithmetic causes a considerable loss of information about the null space of the constraint gradients when $\nabla^{2}p(x)$ is formed. This loss of information renders ordinary trust region Newton's methods unstable and degrades the accuracy of the solution to the trust region problem. The algorithm of More and Sorenson [1983] is modified so as to be more stable and less sensitive to the nature of finite precision arithmetic in this situation. Numerical experiments clearly demonstrate the stability of the proposed algorithm.