  We present the first hybrid measurement of the average muon number in air
showers at ultra-high energies, initiated by cosmic rays with zenith angles
between $62^\circ$ and $80^\circ$. The measurement is based on 174 hybrid
events recorded simultaneously with the Surface Detector array and the
Fluorescence Detector of the Pierre Auger Observatory. The muon number for each
shower is derived by scaling a simulated reference profile of the lateral muon
density distribution at the ground until it fits the data. A $10^{19}$~eV
shower with a zenith angle of $67^\circ$, which arrives at the Surface Detector
array at an altitude of 1450 m above sea level, contains on average $(2.68 \pm
0.04 \pm 0.48\,\mathrm{(sys.)}) \times 10^{7}$ muons with energies larger than
0.3 GeV. The logarithmic gain $\mathrm{d}\ln{N_\mu} / \mathrm{d}\ln{E}$ of
muons with increasing energy between $4\times 10^{18}$ eV and $5 \times
10^{19}$ eV is measured to be $(1.029\, \pm\, 0.024\, \pm
0.030\,\mathrm{(sys.)})$.
