We present the results of a 50 ks long X-ray observation of the relativistic double pulsar system PSR J0737-3039 obtained with the XMM-Newton satellite in 2004 March. The source has a soft spectrum (with a power-law photon index of 3.5+0.5-0.3 ) and a 0.2-10 keV luminosity of ~3×1030 ergs s-1 (assuming a distance of 500 pc), consistent with the values derived from a previous Chandra observation. No flux variations have been detected, implying the absence of large orbital modulations. The high time resolution of the European Photon Imaging Camera has allowed us to perform the first search for X-ray pulsations from this system. The result was negative, with an upper limit of 60% on the pulsed fraction of the 22 ms pulsar.