A search for a Higgs boson produced via vector-boson fusion and decaying into invisible particles is presented, using 20.3 fb$^{-1}$ of proton--proton collision data at a centre-of-mass energy of 8 TeV recorded by the ATLAS detector at the LHC. For a Higgs boson with a mass of 125 GeV, assuming the Standard Model production cross section, an upper bound of 0.28 is set on the branching fraction of $H\to$ invisible at 95% confidence level, where the expected upper limit is 0.31. The results are interpreted in models of Higgs-portal dark matter where the branching fraction limit is converted into upper bounds on the dark-matter--nucleon scattering cross section as a function of the dark-matter particle mass, and compared to results from the direct dark-matter detection experiments.