  The Muon Ionization Cooling Experiment (MICE) collaboration seeks to
demonstrate the feasibility of ionization cooling, the technique by which it is
proposed to cool the muon beam at a future neutrino factory or muon collider.
The emittance is measured from an ensemble of muons assembled from those that
pass through the experiment. A pure muon ensemble is selected using a
particle-identification system that can reject efficiently both pions and
electrons. The position and momentum of each muon are measured using a
high-precision scintillating-fibre tracker in a 4\,T solenoidal magnetic field.
This paper presents the techniques used to reconstruct the phase-space
distributions and reports the first particle-by-particle measurement of the
emittance of the MICE Muon Beam as a function of muon-beam momentum.
