Using recently calculated analytic models for the thermal structure of ultramagnetized neutron stars, we estimate the thermal fluxes from young (t similar to 1000 yr) ultramagnetized (B similar to 10(15) G) cooling neutron stars. We find that the pulsed X-ray emission from objects such as 1E 1841-045 and 1E 2259+586, as well as many soft-gamma repeaters, can be explained by photon cooling if the neutron star possesses a thin insulating envelope of matter of low atomic weight at densities rho<10(7)-10(8) g cm(-3). The total mass of this insulating layer is M similar to 10(-11)-10(-8) M..