  What was the nature of the Late Hesperian climate? Warm and wet or cold and
dry? Formulated this way the question leads to an apparent paradox since both
options seem implausible. A warm and wet climate would have produced extensive
fluvial erosion but few valley networks have been observed at the age of the
late Hesperian. A too cold climate would have kept any northern ocean frozen
most of the time. A moderate cold climate would have transferred the water from
the ocean to the land in the form of snow and ice. But this would prevent
tsunami formation, for which there is some evidence. Here, we provide new
insights from numerical climate simulations in agreement with surface
geological features to demonstrate that the Martian climate could have been
both cold and wet. Using an advanced General Circulation Model (GCM), we
demonstrate that an ocean can be stable, even if the Martian mean surface
temperature is lower than 0$^\circ$C. Rainfall is moderate near the shorelines
and in the ocean. The southern plateau is mostly covered by ice with a mean
temperature below 0$^\circ$C and a glacier return flow back to the ocean. This
climate is achieved with a 1 bar CO$_2$ dominated atmosphere with 10\% H$_2$.
Under this scenario 3 Ga, the geologic evidence of a shoreline and tsunami
deposits along the ocean/land dichotomy are compatible with ice sheets and
glacial valleys in the southern highlands.
