This Letter reports the observation of WZγ production and a measurement of its cross section using 140.1±1.2  fb⁻¹ of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. The WZγ production cross section, with both the W and Z bosons decaying leptonically, pp→WZγ→ℓ′^±νℓ⁺ℓ⁻γ (ℓ^(′)=e, μ), is measured in a fiducial phase-space region defined such that the leptons and the photon have high transverse momentum and the photon is isolated. The cross section is found to be 2.01±0.30(stat)±0.16(syst)  fb. The corresponding standard model predicted cross section calculated at next-to-leading order in perturbative quantum chromodynamics and at leading order in the electroweak coupling constant is 1.50±0.06  fb. The observed significance of the WZγ signal is 6.3σ, compared with an expected significance of 5.0σ.