  A detailed investigation of Ba$_2$MnGe$_2$O$_7$ 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 $T_N \approx 4$ K were determined
using unpolarized neutron diffraction. This data implies no structural phase
transition from 10 K down to 2.5 K and is well described within the tetragonal
space group $P\bar{4}2_1m$. We found that in zero magnetic field the magnetic
space group is either $C_cmc2_1$ or $P_c2_12_12_1$ with antiferromagnetic order
along the [110] or [100] directions, respectively, while neighboring spins
along the [001] axis are ordered antiferromagnetically. A non-collinear 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)
$\mu_B$/Mn$^{2+}$ 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 Ba$_2$MnGe$_2$O$_7$ as compared to those of
Ba$_2$CoGe$_2$O$_7$ and Ca$_2$CoSi$_2$O$_7$ are discussed.
