A measurement of the polarisation of W bosons produced in top-quark decays is presented, using proton–proton collision data at a centre-of-mass energy of s=13 TeV. The data were collected by the ATLAS detector at the Large Hadron Collider and correspond to an integrated luminosity of 139 fb⁻¹. The measurement is performed selecting tt¯ events decaying into final states with two charged leptons (electrons or muons) and at least two b-tagged jets. The polarisation is extracted from the differential cross-section distribution of the cos⁡θ⁎ variable, where θ⁎ is the angle between the momentum direction of the charged lepton from the W boson decay and the reversed momentum direction of the b-quark from the top-quark decay, both calculated in the W boson rest frame. Parton-level results, corrected for the detector acceptance and resolution, are presented for the cos⁡θ⁎ angle. The measured fractions of longitudinal, left- and right-handed polarisation states are found to be f0=0.684±0.005(stat.)±0.014(syst.), fL=0.318±0.003(stat.)±0.008(syst.) and fR=−0.002±0.002(stat.)±0.014(syst.), in agreement with the Standard Model prediction.