The thermal emission of strongly magnetized neutron-star atmospheres is thought to be highly polarized. However, because of the different orientations of the magnetic field over the surface of the neutron star (NS), it is commonly assumed that the net observed polarization will be significantly reduced as the polarization from different regions will cancel each other. We show that the birefringence of the magnetized QED vacuum decouples the polarization modes in the magnetosphere; therefore, the direction of the polarization follows the direction of the magnetic field up to a large distance from the stellar surface. At this distance, the rays that leave the surface and are destined for our detectors pass through only a small solid angle; consequently, the polarization direction of the emission originating in different regions will tend to align together. The net observed polarization of the thermal radiation of NSs should therefore be very large. Measurement of this polarization will be the first direct evidence of the birefringence of the magnetized vacuum due to QED and a direct probe of behavior of the vacuum at magnetic fields of order of and above the critical QED field of 4.4×1013 G. The large observable polarization will also help us learn more about the atmospheric properties of NSs.