  We present new near-infrared VLTI/GRAVITY interferometric spectra that
spatially resolve the broad Br$\gamma$ 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 degree per baseline across the spectrum, we detect a
differential phase signal that reaches a maximum of ~0.5 degree between the
line and continuum. This represents an offset of ~120 $\mu$as (0.14 pc) between
the BLR and the centroid of the hot dust distribution traced by the 2.3 $\mu$m
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 Br$\gamma$ line. We infer the
radius of the BLR to be ~65 $\mu$as (0.075 pc), which is consistent with the
radius-luminosity relation of nearby active galactic nuclei derived based on
the time lag of the H$\beta$ line from reverberation mapping campaigns. Our
dynamical modelling indicates the black hole mass is $\sim
1\times10^8\,M_\odot$, which is a little below, but consistent with, the
standard $M_{\rm BH}$-$\sigma_*$ relation.
