We present the first hybrid measurement of the average muon number in air showers at ultrahigh energies, initiated by cosmic rays with zenith angles between 62° and 80°. 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°, which arrives at the surface detector array at an altitude of 1450 m above sea level, contains on average (2.68±0.04±0.48(sys))×10^7 muons with energies larger than 0.3 GeV. The logarithmic gain dlnNμ/dlnE of muons with increasing energy between 4×10^18 eV and 5×10^19 eV is measured to be (1.029±0.024±0.030(sys)).
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