Understanding the doping mechanisms in the simplest superconducting copper oxide - the infinite-layer compound ACuO₂ (where A is an alkaline earth metal) - is an excellent way of investigating the pairing mechanism in high-transition-temperature (high-T_c) superconductors more generally^1-4. Gate-induced modulation of the carrier concentration^5-7 to obtain superconductivity is a powerful means of achieving such understanding: It minimizes the effects of potential scattering by impurities, and of structural modifications arising from chemical dopants. Here we report the transport properties of thin films of the infinite-layer compound CaCuO₂ using field-effect doping. At high hole- and electron-doping levels, superconductivity is induced in the nominally insulating material. Maximum values of T_c of 89 K and 34 K are observed respectively for hole- and electron-type doping of around 0.15 charge carriers per CuO₂. We can explore the whole doping diagram of the CuO₂ plane while changing only a single electric parameter, the gate voltage.