Graphene has demonstrated great promise for future electronics technology as well as fundamental physics applications because of its linear energy-momentum dispersion relations which cross at the Dirac point. However, accessing the physics of the low-density region at the Dirac point has been difficult because of disorder that leaves the graphene with local microscopic electron and hole puddles. Efforts have been made to reduce the disorder by suspending graphene, leading to fabrication challenges and delicate devices which make local spectroscopic measurements difficult. Recently, it has been shown that placing graphene on hexagonal boron nitride (hBN) yields improved device performance. Here we use scanning tunnelling microscopy to show that graphene conforms to hBN, as evidenced by the presence of Moire patterns. However, contrary to predictions, this conformation does not lead to a sizeable band gap because of the misalignment of the lattices. Moreover, local spectroscopy measurements demonstrate that the electron-hole charge fluctuations are reduced by two orders of magnitude as compared with those on silicon oxide. This leads to charge fluctuations that are as small as in suspended graphene, opening up Dirac point physics to more diverse experiments.