When a liquid is cooled to produce a glass its dynamics, dominated by the structural relaxation, become very slow, and at the glass-transition temperature Tg its characteristic relaxation time is about 100 s. At slightly elevated temperatures (~1.2 Tg) however, a second process known as the Johari-Goldstein relaxation, βJG, decouples from the structural one and remains much faster than it down to Tg. While it is known that the βJG-process is strongly coupled to the structural relaxation, its dedicated role in the glass-transition remains under debate. Here we use an experimental technique that permits us to investigate the spatial and temporal properties of the βJG relaxation, and give evidence that the molecules participating in it are highly mobile and spatially connected in a system-spanning, percolating cluster. This correlation of structural and dynamical properties provides strong experimental support for a picture, drawn from theoretical studies, of an intermittent mosaic structure in the deeply supercooled liquid phase.