A search for the decay of a light Higgs boson (120--140 GeV) to a pair of weakly interacting, long-lived particles in $1.94\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton-proton collisions at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$ recorded in 2011 by the ATLAS detector is presented. The search strategy requires that both long-lived particles decay inside the muon spectrometer. No excess of events is observed above the expected background and limits on the Higgs boson production times branching ratio to weakly interacting, long-lived particles are derived as a function of the particle proper decay length.