  We have performed saturated absorption spectroscopy on the cesium D$_{2}$
line in 3 T and 7 T magnetic fields. By means of sideband spectroscopy on the
extreme angular momentum states we have measured the linear magnetic frequency
shift of the transition to be $\gamma_1 = 13.994\:301(11)$ GHz/T. This
corresponds to an optical magnetic field measurement of better than 1 ppm
accuracy. From this value we can calculate the fine structure Land\'e g-factor
$g_J\left(6^2P_{3/2}\right) = 1.334\:087\:49(52)$. This is consistent with the
previous best measurement, and improves the accuracy by more than two orders of
magnitude. We have also measured, for the first time ever, the quadratic
diamagnetic shift as $\gamma_2 = 0.4644(35)\:\mathrm{MHz/T^2}$. Our work opens
up the field of accurate high field optical magnetometry using atomic cesium.
These high accuracy measurements also allow for testing of advanced atomic
structure models, as our results are incompatible with the Russel-Saunders
coupling value, and the hydrogen-constant-core-model value, by 31 and 7
standard deviations respectively.
