Two new techniques have been developed for making high-accuracy measurements of the thermal conductivity kappa in chemical-vapor-deposited (CVD) diamond films: (1) a steady state technique for measuring kappa(parallel-to) (heat flowing in a direction parallel.to the plane of the sample) and (2) a laser flash technique for measuring kappa(perpendicular-to) (heat flowing perpendicular to the plane of the film). By measuring kappa(parallel-to) and kappa(perpendicular-to) for a series of high-quality CVD diamond films of different thicknesses, we are able to extract local values for these variables as a function of height z above the substrate surface. Both show a large gradient with respect to z, with kappa(local) rising more rapidly with z than kappa(parallel-to)local for approximately the first 200 mum. This is consistent with phonon scattering from impurities and defects if they are preferentially located near grain boundaries of the columnar structure. For z less than or similar to 300 mum, the local conductivity is nearly isotropic with the very high value of 23-24 W cm-1-degrees-C-1 at 25-degrees-C, to be compared with 22 W cm-1-degrees-C-1 for the best type IIa single-crystal diamond so far reported. These results have direct implications for the thermal management of microelectronic devices if the remarkable conductivity of diamond is to be used most effectively.