Deep optical B band images of the ROSAT HRI error region of RX J0720.4-3125 reveal the presence of two faint stellar-like objects with B = 26.1 +/- 0.25 and B = 26.5 +/- 0.30. Exposures obtained through U, V and I filters are not sensitive enough to detect the two candidates and provide upper limits of U = 24.9, V = 23.2 and I = 21.9. These new observations virtually establish that RX J0720.4-3125 is a slowly rotating, probably completely isolated neutron star. The absence of an optical counterpart brighter than B = 26.1 seems incompatible with a neutron star atmosphere having a chemical composition dominated by Hydrogen or Helium. UBI photometry of field stars shows astonishingly little interstellar reddening in the direction of the X-ray source. Together with the small column density detected by the ROSAT PSPC, this suggests a mean particle density in the range of n = 0.1 - 0.4 cm-3. Such average densities would imply very low velocities relative to interstellar medium (Vrel < 10 km/s) if the source were powered by accretion. These stringent constraints may be relaxed if the neutron star is presently crossing a small size structure of higher density or if the effective temperature of the heated atmosphere is overestimated by the blackbody approximation. Alternatively, RX J0720.4-3125 could be a young and highly magnetized cooling neutron star.
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