  We report on a 50 ksec XMM-Newton observation of the double pulsar system
J0737-3039 performed on April 2004. We present results of the spectral analysis
of these data combined with the much shorter Chandra pointing performed on
January 2004. Black body emission with effective temperature of
0.20^{+0.02}_{-0.02} keV (90% confidence level) and emission radius
75^{+30}_{-9} m for a distance of 0.5 kpc (implying a 0.5-10 keV luminosity
\~6x10^{29} erg/s) is a viable interpretation, calling for a stream of
particles accelerated in the magnetosphere of PSR J0737-3039A and depositing
their kinetic energy in the magnetic polar cap of PSR J0737-3039A or of the
companion PSR J0737-3039B. A single power-law emission model implies a very
steep photon index Gamma=4.2^{+2.1}_{-1.2} and a suspiciously high hydrogen
column density, whereas a photon index Gamma=2 does not provide an adequate
description of the XMM-Newton and Chandra data. A two component model (a black
body plus a power-law with Gamma=2) is statistically acceptable, but the
additional power-law component is not required by the data.
