In order to get detailed insights into magnetism and its relationship with superconductivity, neutron scattering experiments were carried out on high-quality single crystals of the non-centrosymmetric heavy-fermion superconductor CePt3Si, with T-N=2.2 K and T-SC similar to 45 K. All the samples used in this study were characterised via specific heat measurements. Sharp specific heat jumps at T-sc=0.45 K, and T-N=2.2K for all samples prove the high quality of the crystal. The neutron diffraction study confirms that an antiferromagnetic peak described with the propagation vector q=(0 0 1/2) persists inside the superconducting state, with a tiny suppression of the magnetic intensity below T-sc. In contrast, an external magnetic field along the [0 0 1] direction causes a drastic change in the antiferromagnetic state; the antiferromagnetic intensity shows an enormous enhancement with increasing applied field. The magnetic intensity does not show any anomaly at B-c2, instead there exists a small inflection at around B' similar to 3.8 T where an additional phase boundary was reported. The huge increase in antiferromagnetic intensity corresponds to an increase of the staggered moment under uniform field, opposite of what is observed in ordinary antiferromagnets. These features suggest a weak coupling between magnetic and superconducting order parameters, and clearly evidence the anomalous antiferromagnetism in this compound.