Recent Hubble Space Telescope (HST)/Advanced Camera for Surveys (ACS) observations revealed the presence of a double subgiant branch (SGB) in the core of the Galactic globular cluster NGC 1851. This peculiarity was tentatively explained by the presence of a second population with either an age difference of about 1 Gyr, or a higher C+N+O abundance, probably due to pollution by the first generation of stars. In this Letter, we analyze VLT-FORS V, I images, covering 12farcm7 × 12farcm7, in the southwest quadrant of the cluster, allowing us to probe the extent of the double SGB from ~1farcm4 to ~13' from the cluster center. Our study reveals, for the first time, that the "peculiar" population is the one associated with the fainter SGB. Indeed, while the percentage of stars in this sequence is about 45% in the cluster core (as previously found on the basis of ACS/HST data), we find that it drops sharply, to a level consistent with zero in our data, at ~2farcm4 from the cluster center, where the brighter SGB, in our sample, still contains ~100 stars. Implications for the proposed scenarios are discussed. Observations collected at the European Southern Observatory, Paranal, Chile (ESO program 68.D-0510A).