We describe 2004 May observations of the 22.7-ms ‘A’ pulsar in the double pulsar binary system, J0737–3039. Our data were obtained with a coherent dedispersion system at 20 and 50 cm wavelength bands, during an intensive 15 d observing session at the Parkes radio telescope. High signal-to-noise-ratio (S/N) polarimetric profiles of the ‘A’ pulsar are presented; these profiles provide templates against which to search for evolution of the pulse profile in coming years. We measure flux densities for the ‘A’ pulsar of 1.2 ± 0.1 mJy at 1373 MHz and 4.2 ± 0.5 mJy at 685 MHz. Faraday rotation is also detected in both bands; the rotation measure of the pulsar is −121 ± 4 rad m−2 at 1373 MHz and −118.4 ± 0.3 rad m−2 at 685 MHz, implying a mean longitudinal component of the magnetic field of approximately 3 μG along the line of sight to the pulsar. Pulse arrival times from individual 2-min integrations are analysed and we detect a clear signature of Shapiro delay in the timing residuals. Assuming a mass of 1.25 M⊙ for the companion ‘B’ pulsar, we derive a limit of 88.5+0.8 °−1.1 on the inclination angle of the system, almost consistent with the most recent limit of 0.29 ± 0.14° away from 90°, obtained from scintillation studies. The need to obtain many arrival times near inferior conjunction forces us to use low S/N profiles for our analysis. This can lead to unrealistic error estimates if standard timing techniques are used. In an appendix, we describe how to obtain more realistic error estimates and how to avoid some catastrophic consequences of building a standard template profile from low S/N data.