We present new near-infrared VLTI/GRAVITY interferometric spectra that spatially resolve the broad Brγ emission line in the nucleus of the active galaxy IRAS 09149-6206. We use these data to measure the size of the broad line region (BLR) and estimate the mass of the central black hole. Using an improved phase calibration method that reduces the differential phase uncertainty to 0.05° per baseline across the spectrum, we detect a differential phase signal that reaches a maximum of ∼0.5° between the line and continuum. This represents an offset of ∼120 μas (0.14 pc) between the BLR and the centroid of the hot dust distribution traced by the 2.3 μm continuum. The offset is well within the dust sublimation region, which matches the measured ∼0.6 mas (0.7 pc) diameter of the continuum. A clear velocity gradient, almost perpendicular to the offset, is traced by the reconstructed photocentres of the spectral channels of the Brγ line. We infer the radius of the BLR to be ∼65 μas (0.075 pc), which is consistent with the radius-luminosity relation of nearby active galactic nuclei derived based on the time lag of the Hβ line from reverberation mapping campaigns. Our dynamical modelling indicates the black hole mass is ∼1 × 108 M☉, which is a little below, but consistent with, the standard MBH-σ* relation.