We consider the successive measurement of position and momentum of a single particle. Let P be the conditional probability to measure the momentum k with precision \Delta k, given a previously successful position measurement q with precision \Delta q. Several upper bounds for the probability P are derived. For arbitrary, but given precisions \Delta q and \Delta k, these bounds refer to the variation of q, k, and the state vector \psi of the particle. The first bound is given by the inequality {P}\leq \Delta k\Delta q/h, where h is Planck's quantum of action. It is nontrivial for all measurements with \Delta k\Delta q less then h. A sharper bound is obtained by applying the {Hilbert--Schmidt} norm. As our main result, the {least upper bound} of P is determined. All bounds are independent of the order with which the measuring of the position and momentum is made.

PACS numbers: 03.65.Ta, 04.80.Nn, 03.67.--a