Longitudinal magnetic field determinations have been performed for a sample of Ap stars, from CCD spectra recorded simultaneously in both circular polarizations using the Zeeman analyzer of the ESO Cassegrain Echelle Spectrograph fed by the ESO 3.6 m telescope. The magnetic fields have been derived by the classical "photographic" method, i.e. by measuring the wavelength shift of lines between both polarizations. Comparison with previous studies demonstrate the proper operation of the Zeeman analyzer, which was used for the first time. The accuracy of the longitudinal field determination that is achieved in the best cases is of the order of 80 G. It is limited by the spectral resolution of the observations and consistent with theoretical estimates. Most curves of rotational variation of the longitudinal field are found to be closely sinusoidal, confirming that the anharmonicities of these curves that had been reported in the past were often due to pitfalls of the measurements of the photographic plates, which are avoided with the present, low-noise, CCD spectra. Nevertheless, two of the studied stars (HD 119419 and HD 175362) appear to have definitely nonsinusoidal longitudinal field variation curves, probably indicative of a magnetic geometry somewhat more complex than a simple dipole.