We used the high-resolution Q3D magnetic spectrograph at the Maier-Leibnitz Laboratory (MLL) Tandem accelerator in Munich to study 200Hg and 202Hg after two-neutron transfer. The results confirm the sharp drop in the number of low-lying 0+ states towards the 208Pb shell closure. In total, we assigned six 0+ states in 200Hg and four 0+ states in 202Hg. The 0+ excitation energies and the measured (p,t) transfer cross sections indicate a structural change throughout the Hg isotopes, with the most notable result being the peaking in the cross section of the 02+ state in 200Hg.