Using XMM-Newton and Chandra, we achieved phase-connected timing of the 105 ms X-ray pulsar PSR J1852+0040 that provides the first measurement of the spin-down rate of a member of the class of Central Compact Objects (CCOs) in supernova remnants. We measure P-dot = 8.68(9)E-18, and find no evidence for timing noise or variations in X-ray flux over 4.8 yr. In the dipole spin-down formalism, this implies a surface magnetic field strength B_s = 3.1E10 G, the smallest ever measured for a young neutron star, and consistent with being a fossil field. In combination with upper limits on B_s from other CCO pulsars, this is strong evidence in favor of the "anti-magnetar" explanation for their low luminosity and lack of magnetospheric activity or synchrotron nebulae. While this dipole field is small, it can prevent accretion of sufficient fall-back material so that the observed X-ray luminosity of L_x = 5.3E33(d/7.1 kpc)^2 erg/s must instead be residual cooling. The spin-down luminosity of PSR J1852+0040, E-dot = 3.0E32 erg/s, is an order-of-magnitude smaller than L_x. Fitting of the X-ray spectrum to two blackbodies finds small emitting radii, R_1 = 1.9 km and R_2 = 0.45 km, for components of kT_1 = 0.30 keV and kT_2 = 0.52 keV, respectively. Such small, hot regions are ubiquitous among CCOs, and are not yet understood in the context of the anti-magnetar picture because anisotropic surface temperature is usually attributed to the effects of strong magnetic fields. ; Comment: 11 pages, 7 figures, Added text and figures, acccepted by The Astrophysical Journal