A detailed investigation of Ba2MnGe2O7 was performed in its low-temperature magnetoelectric state combining neutron diffraction with magnetization measurements on single crystals. In the paramagnetic state at 10 K, polarized neutron diffraction was applied to reveal the components of the susceptibility tensor. The crystal and magnetic structures below the antiferromagnetic transition temperature of TN approximate to 4 K were determined using unpolarized neutron diffraction. These data imply no structural phase transition from 10 K down to 2.5 K and are well described within the tetragonal space group P4 over bar 21m. We found that in zero magnetic field the magnetic space group is either Ccmc21 or Pc212121 with antiferromagnetic order along the [110] or [100] direction, respectively, while neighboring spins along the [001] axis are ordered antiferromagnetically. A noncollinear spin arrangement due to small canting within the ab plane is allowed by symmetry and observed experimentally. The ordered moment is found to be 3.24(3) mu B/Mn2+ at 2.5 K and the temperature-field dependent magnetic phase diagram is mapped out by macroscopic magnetization. Distinct differences between the magnetic structure of Ba2MnGe2O7 as compared to those of Ba2CoGe2O7 and Ca2CoSi2O7 are discussed.