  Many extrasolar planets orbit closely to their parent star. Their existence
raises the fundamental problem of loss and gain in their mass. For exoplanet
HD209458b, reports on an unusually extended hydrogen corona and a hot layer in
the lower atmosphere seem to support the scenario of atmospheric inflation by
the strong stellar irradiation. However, difficulties in reconciling
evaporation models with observations call for a reassessment of the problem.
Here, we use HST archive data to report a new absorption rate of ~8.9% +/- 2.1%
by atomic hydrogen during the HD209458b transit, and show that no sign of
evaporation could be detected for the exoplanet. We also report evidence of
time variability in the HD209458 Lyman-a flux, a variability that was not
accounted for in previous studies, which corrupted their diagnostics. Mass loss
rates thus far proposed in the literature in the range 5x(10^{10}-10^{11} g
s^{-1}) must induce a spectral signature in the Lyman-a line profile of
HD209458 that cannot be found in the present analysis. Either an unknown
compensation effect is hiding the expected spectral feature or else the mass
loss rate of neutrals from HD209458 is modest.
