The NA62 experiment at the CERN SPS will measure the branching ratio for the decay K+ → π+ v- to within ≈10%. The large-angle veto (LAV) detectors must detect photons with energies as low as 200 MeV with an inefficiency of less than 10-4. After a comprehensive series of comparative studies, NA62 decided to base the LAV system on the lead-glass block/PMT assemblies recycled from the OPAL barrel calorimeter, and a prototype veto station, complete with front-end electronics, was constructed and tested in 2009. Eleven additional stations are to be constructed before data taking begins in 2012. We describe the design, construction, and testing of the LAV system, including the recovery and mechanical adaptation of the OPAL hardware, issues related to operation in high vacuum, HV distribution, and in-situ monitoring, and especially, the design of an efficient and economical system for the readout of time and energy over an extended dynamic range.