The inclusive cross-section for the production of a single top quark in association with a $W$ boson is measured using 140 fb$^{-1}$ of proton-proton collision data collected with the ATLAS detector at $\sqrt{s} = 13$ TeV. Events containing two charged leptons and at least one jet identified as originating from a $b$-quark are selected. A multivariate discriminant is constructed to separate the $tW$ signal from the $t\bar{t}$ background. The cross-section is extracted using a profile likelihood fit to the signal and control regions and it is measured to be $\sigma_{tW} = 75^{+15}_{-14}$ pb, in good agreement with the Standard Model prediction. The measured cross-section is used to extract a value for the left-handed form factor at the $Wtb$ vertex times the CKM matrix element $|f_\text{LV}V_{tb}|$ of $0.97 \pm 0.10$.