Black carbon (BC) aerosol absorbs sunlight that might have otherwise been reflected to space and changes the radiative heating of the atmosphere and surface. These effects may alter the dynamical and hydrological processes governing cloud formation. A new, microphysical, effect of BC on climate is identified here, in which solar heating within BC-containing cloud condensation nuclei (CCN) slows or prevents the activation of these CCN into cloud drops. Solar-heated BC-containing droplets are elevated in temperature by fractions of a degree above the ambient, thus raising the droplet vapor pressure and inhibiting activation of the most absorptive CCN. This paper develops the theory describing the alteration of the K¿hler curve (i.e., the equilibrium vapor pressure over a droplet as a function of water uptake) as a function of CCN size and BC fraction. The effect is most significant in those CCN that contain volumes of BC larger than a 500 nm diameter sphere. For an aerosol population with 10% BC mass fraction per particle, solar heating can cause a 10% reduction in the CCN concentration at 0.01% critical supersaturation. On the other hand, the effect of heating by BC absorption on CCN activation above ~0.1% critical supersaturation is negligible.