We discuss the prospect of detecting a gravitational wave background by means of a simultaneous Doppler tracking of two spacecraft. We find that the cross spectrum of the Doppler shifts of the two spacecraft is a filtered expression of the energy density spectrum of the background. The filter function, which is expressed as a series in terms of Legendre polynomials, is obtained by an integration over the rotation group, assuming the background to be isotropic. We examine the main noise sources and point out the advantages of a measurement with two spacecraft.