A direct search for evidence of decays to invisible particles of a Higgs boson at the Large Hadron Collider (LHC) is presented. This search is performed for a Standard Model (SM)-like Higgs boson produced in association with a boson and having a mass between 115 GeV and 300 GeV. The results are interpreted to place limits on the branching fraction to invisible particles of the SM Higgs boson, where a non-zero invisible branching fraction could provide evidence for the production of dark matter particles. In addition, limits are set on any neutral Higgs-like particle, produced in association with a boson and decaying predominantly to invisible particles. No deviation from the Standard Model expectation is observed in the search, which uses 4.7 fb of 7 TeV collision data and 13.0 fb of 8 TeV collision data collected by the ATLAS experiment at the LHC. Assuming the production rate for a 125 GeV SM Higgs boson, an invisible branching fraction greater than 65% is excluded with a 95% confidence level for the observed, and 84% with 95% CL for the expected. Limits are also set on the cross section times invisible branching fraction of a possible additional Higgs-like boson over the mass range 115 GeV <<300 GeV.