Single top-quark production in association with a Z boson, where the Z boson decays to a pair of charged leptons, is measured in the trilepton channel. The proton-proton collision data collected by the ATLAS experiment from 2015 to 2018 at a centre-of-mass energy of 13 TeV are used, corresponding to an integrated luminosity of 139 fb$^{−1}$. Events containing three isolated charged leptons (electrons or muons) and two or three jets, one of which is identified as containing a b-hadron, are selected. The main backgrounds are from $ t\overline{t}Z $ and diboson production. Neural networks are used to improve the background rejection and extract the signal. The measured cross-section for tℓ$^{+}$ℓ$^{−}$q production, including non-resonant dilepton pairs with $ {m}_{{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-}} $> 30 GeV, is 97 ± 13 (stat.) ± 7 (syst.) fb, consistent with the Standard Model prediction.