We report on the fabrication and measurement of nanoscale devices that permit electrostatic confinement in bilayer graphene on a substrate. The graphene bilayer is sandwiched between hexagonal boron nitride bottom and top gate dielectrics. Top gates are patterned such that constrictions and islands can be electrostatically induced. The high quality of the devices becomes apparent from the smooth pinch-off characteristics of the constrictions at low temperature with features indicative of conductance quantization. The islands exhibit clear Coulomb blockade and single-electron transport.