An observation of vector-boson-fusion (VBF) production of Higgs bosons in the H{\rightarrow\,}WW^{\ast}{\rightarrow\,}e\nu\mu\nu decay channel is presented. The proton-proton collision data used to obtain this result were produced at the LHC with a center-of-mass energy of 13 TeV and recorded with the ATLAS detector between 2015 and 2018, corresponding to an integrated luminosity of \mathrm{139~fb^{-1}}. The signal is observed (expected) with a significance of 7.0 (6.2) standard deviations. The product of the total VBF cross-section times the {H{\rightarrow\,}WW^{\ast}} branching fraction is measured to be 0.85 \pm 0.10\;(\mathrm{stat.})\;^{+0.17}_{-0.13}\;(\mathrm{syst.})~\mathrm{pb}, compatible with the Standard Model prediction of $0.81\pm 0.02$ pb.

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