HR8799 is orbited by at least four giant planets, making it a prime target for the recently commissioned Spectro-Polarimetric High-contrast Exoplanet REsearch (VLT/SPHERE). As such, it was observed on five consecutive nights during the SPHERE science verification in December 2014. We aim to take full advantage of the SPHERE capabilities to derive accurate astrometric measurements based on H-band images acquired with the Infra-Red Dual-band Imaging and Spectroscopy (IRDIS) subsystem, and to explore the ultimate astrometric performance of SPHERE in this observing mode. We also aim to present a detailed analysis of the orbital parameters for the four planets. We report the astrometric positions for epoch 2014.93 with an accuracy down to 2.0 mas, mainly limited by the astrometric calibration of IRDIS. For each planet, we derive the posterior probability density functions for the six Keplerian elements and identify sets of highly probable orbits. For planet d, there is clear evidence for nonzero eccentricity ($e \simeq 0.35$), without completely excluding solutions with smaller eccentricities. The three other planets are consistent with circular orbits, although their probability distributions spread beyond $e = 0.2$, and show a peak at $e \simeq 0.1$ for planet e. The four planets have consistent inclinations of about $30\deg$ with respect to the sky plane, but the confidence intervals for the longitude of ascending node are disjoint for planets b and c, and we find tentative evidence for non-coplanarity between planets b and c at the $2 \sigma$ level.