Dimensionality plays a fundamental role in cuprate high-temperature superconductivity; all high-temperature cuprate superconductors adopt a layered crystal structure, and much of high-temperature superconductivity theory is based on purely two-dimensional (2D) models. A monolayer Bi2201 contains only a single layer of CuO₂ plane, and therefore represents a cuprate superconductor in the ultimate 2D limit. Here, we exfoliate La doped Bi2201 (Bi₂Sr_2-xLa_xCuO_6+δ) single crystals down to monolayer (i.e. half unit cell) for transport measurements. We are able to tune the doping level of Bi2201 by annealing the sample in vacuum or ozone, and study the strongly correlated electronic states in Bi2201 over the entire phase diagram in a single sample. In addition, we probe the electronic structure of monolayer Bi2201 at atomic scale using scanning tunnelling microscopy and spectroscopy.