In the problem considered, a vector of many imprecise measurements (e.g., spectroscopic) is used to linearly predict a quantity whose precise measurement is difficult or expensive. The regression vector is estimated from a calibration experiment having both types of measurements for a random sample. Most previous approaches to this problem adjust for approximate multicollinearity, which often results from the correlations among the imprecise measurements, by inverting an approximation (e.g., factor-analytic) to their covariance matrix. In the approach here, it is argued that the regression vector should lie in a lower dimensional suhspace determined by the principal components of the covariance matrix. It is then estimated by applying a Stein contraction of the least squares estimator to the principal components regression estimator. Examples using real data are presented in which the proposed estimator substantially improves on the ordinary least squares estimation.