We have detected longitudinal magnetic fields between 2 and 4 kG in three (WD 0446$-$790, WD 1105$-$048, WD 2359$-$434) out of a sample of 12 normal DA white dwarfs by using optical spectropolarimetry done with the VLT Antu 8 m telescope equipped with FORS1. With the exception of 40 Eri B (4 kG) these are the first positive detections of magnetic fields in white dwarfs below 30 kG. Although suspected, it was not clear whether a significant fraction of white dwarfs contain magnetic fields at this level. These fields may be explained as fossil relics from magnetic fields in the main-sequence progenitors considerably enhanced by magnetic flux conservation during the shrinkage of the core. A detection rate of 25 % (3/12) may indicate now for the first time that a substantial fraction of white dwarfs have a weak magnetic field. This result, if confirmed by future observations, would form a cornerstone for our understanding on the evolution of stellar magnetic fields. Keywords: stars: white dwarfs - stars: magnetic fields - stars: individual: WD0446-790, WD1105-048, WD2359-434