Aims: Observations of shell-type supernova remnants (SNRs) in the GeV tomulti-TeV γ-ray band, coupled with those at millimetre radiowavelengths, are motivated by the search for cosmic-ray accelerators inour Galaxy. The old-age mixed-morphology SNR W 28 (distance ~2 kpc) is aprime target due to its interaction with molecular clouds along itsnortheastern boundary and other clouds situated nearby. Methods:We observed the W 28 field (for ~40 h) at very high energy (VHE)γ-ray energies (E > 0.1 TeV) with the HESS. Cherenkovtelescopes. A reanalysis of EGRET E > 100 MeV data was alsoundertaken. Results from the NANTEN 4 m telescope Galactic plane surveyand other CO observations were used to study molecular clouds. Results: We have discovered VHE γ-ray emission (HESS J1801-233)coincident with the northeastern boundary of W 28 and a complex ofsources (HESS J1800-240A, B and C) ~0.5° south of W 28 in theGalactic disc. The EGRET source (GRO J1801-2320) is centred on HESSJ1801-233 but may also be related to HESS J1800-240 given the largeEGRET point spread function. The VHE differential photon spectra arewell fit by pure power laws with indices Γ ~2.3 to 2.7. Thespectral indices of HESS J1800-240A, B, and C are consistent withinstatistical errors. All VHE sources are ~10' in intrinsic radius exceptfor HESS J1800-240C, which appears pointlike. The NANTEN12CO(J = 1-0) data reveal molecular clouds positionallyassociating with the VHE emission, spanning a ~15 km s-1range in local standard of rest velocity. Conclusions: TheVHE/molecular cloud association could indicate a hadronic origin forHESS J1801-233 and HESS J1800-240, and several cloud components inprojection may contribute to the VHE emission. The clouds havecomponents covering a broad velocity range encompassing the distanceestimates for W 28 (~2 kpc) and extending up to ~4 kpc. Assuminghadronic origin and distances of 2 and 4 kpc for cloud components, therequired cosmic-ray density enhancement factors (with respect to thesolar value) are in the range ~10 to ~30. If situated at 2 kpc distance,such cosmic-ray densities may be supplied by SNRs like W 28.Additionally and/or alternatively, particle acceleration may come fromseveral catalogued SNRs and SNR candidates, the energetic ultra compactHII region W 28A2, and the HII regions M 8 and M 20, along with theirassociated open clusters. Further sub-mm observations would berecommended to probe in detail the dynamics of the molecular clouds atvelocites > 10 km s-1 and their possible connection to W28.