A new structure is proposed for localizing electromagnetic waves and energies with a left-handed-medium (LHM) slab and a perfectly electrically conducting (PEC) plane. When a current source is placed in front of a perfectly matched LHM slab with negative permittivity -epsilon0 and negative permeability -mu0 and a PEC plane is placed at the image point, we show rigorously that all the electromagnetic waves are confined in a region between the source and the PEC plane, and the fields outside the region are completely zero. Such an energy-localization system would be useful in medical treatments that use concentrated optical and microwave energies. However, a perfectly matched LHM is unphysical and does not exist in nature. Hence we further study the loss and retardation effects of LHM on the energy localization. Numerical results are presented for the lossy LHM structure to demonstrate the energy localization.