The ability to confine light is important both scientifically and technologically. Many light confinement methods exist, but they all achieve confinement with materials or systems that forbid outgoing waves. These systems can be implemented by metallic mirrors, by photonic band-gap materials(1), by highly disordered media (Anderson localization(2)) and, for a subset of outgoing waves, by translational symmetry (total internal reflection1) or by rotational or reflection symmetry(3,4). Exceptions to these examples exist only in theoretical proposals(5-8). Here we predict and show experimentally that light can be perfectly confined in a patterned dielectric slab, even though outgoing waves are allowed in the surrounding medium. Technically, this is an observation of an 'embedded eigenvalue' 9-namely, a bound state in a continuum of radiation modes-that is not due to symmetry incompatibility(5-8,10-16). Such a bound state can exist stably in a general class of geometries in which all of its radiation amplitudes vanish simultaneously as a result of destructive interference. This method to trap electromagnetic waves is also applicable to electronic(12) and mechanical waves(14,15).