This paper deals with the optimal estimation of a distributed-parameter process, for which the observation subject to additive noise is realized by a point sensor. The sensor is allowed to move in the spatial domain, driven by a control input. The aim is to find a trajectory or the sensor which minimizes a weighted sum of the control energy and of the variance of the estimation error, assuming that the estimate is a linear functional of the observation. First, the existence of a solution of this problem is proved. Then an approximation scheme is introduced: it enables us to reduce the problem into an optimal control one for a lumped-parameter system. Several convergence properties of the approximation are proved. Finally, a gradient algorithm is proposed for the numerical solution of the problem and the results of an applications example are shown.