A method for the measurement of pulsed magnetic fields, based on the Zeeman effect of the Hg 253.7 nm line, is described. The advantages of using absorption, rather than emission, lines are stressed. The accuracy of measurement is limited mainly by the error in the determination of the separation of the Zeeman $\sigma $ components. This contribution to the error is independent of field, and, for the apparatus described, is equivalent to $\pm $ 100 G (0.04% at 250 kG). Measurements have been made in the range 50 to 330 kG with measurement times of approximately 2 $\mu $s. Calculations are presented to show that the Zeeman splitting of the Hg 253.7 nm line is a linear function of the field to high accuracy. The deviations from linearity are less than one part per million for fields less than 1.4 MG, less than 0.1% below 45 MG, and less than 1% up to 300 MG. The modifications to the apparatus necessary for work at extremely high fields (B > 4 MG) are discussed, and the effect of these changes on the accuracy is considered.