The Cauchy problem (also called the initial value problem) for systems of ordinary differential equations with right-hand sides depending on some unknown parameters is considered here. The noisy measurements of one of the variables at the given time moments are assumed to be known. A new algorithm for recovering (identifying) the model parameters is proposed in this paper. The algorithm is based on numerical integration of the gradient equations of a weighted least-squares functional. The right-hand sides of the gradient equations are obtained by numerical integration of the Cauchy problem for the original equations and the Cauchy problem for their partial derivatives with respect to unknown parameters. Numerical experiments for the well-known Lotka-Volterra model of oscillating chemical reactions demonstrate the robustness of the proposed algorithm when the measurements are corrupted by random multiplicative noise. All computations are performed using MATLAB^® version 6.0.