  We present the first optical observations of the unique system J0737-3039
(composed of two pulsars, hereafter PSR-A and PSR-B). Ultra-deep optical
observations, performed with the High Resolution Camera of the Advanced Camera
for Surveys on board the Hubble Space Telescope could not detect any optical
emission from the system down to m_F435W=27.0 and m_F606W=28.3. The estimated
optical flux limits are used to constrain the three-component (two thermal and
one non-thermal) model recently proposed to reproduce the XMM-Newton X-ray
spectrum. They suggest the presence of a break at low energies in the
non-thermal power law component of PSR-A and are compatible with the expected
black-body emission from the PSR-B surface. The corresponding efficiency of the
optical emission from PSR-A's magnetosphere would be comparable to that of
other Myr-old pulsars, thus suggesting that this parameter may not dramatically
evolve over a time-scale of a few Myr.
