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 $sim 1times10^8,M_odot$, which is a little below, but consistent with, the standard $M_{rm BH}$-$sigma_*$ relation.
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