  We describe the method devised to reconstruct inclined cosmic-ray air showers
with zenith angles greater than $60^\circ$ detected with the surface array of
the Pierre Auger Observatory. The measured signals at the ground level are
fitted to muon density distributions predicted with atmospheric cascade models
to obtain the relative shower size as an overall normalization parameter. The
method is evaluated using simulated showers to test its performance. The energy
of the cosmic rays is calibrated using a sub-sample of events reconstructed
with both the fluorescence and surface array techniques. The reconstruction
method described here provides the basis of complementary analyses including an
independent measurement of the energy spectrum of ultra-high energy cosmic rays
using very inclined events collected by the Pierre Auger Observatory.
