Various thermodynamic equilibrium properties of naturally abundant, hexagonal ice (ice Ih) of water (H2O) have been used to develop a Gibbs energy function g(T,p) of temperature and pressure, covering the ranges 0-273.16 K and 0 Pa-210 MPa, expressed in the temperature scale ITS-90. It serves as a fundamental equation from which additional properties are obtained as partial derivatives by thermodynamic rules. Extending previously developed Gibbs functions, it covers the entire existence region of ice Ih in the T-p diagram. Close to zero temperature, it obeys the theoretical cubic limiting law of Debye for heat capacity and Pauling's residual entropy. It is based on a significantly enlarged experimental data set compared to its predecessors. Due to the inherent thermodynamic cross relations, the formulas for particular quantities like density, thermal expansion, or compressibility are thus fully consistent with each other, are more reliable now, and extended in their ranges of validity. In conjunction with the IAPWS-95 formulation for the fluid phases of water, the new chemical potential of ice allows an alternative computation of the melting and sublimation curves, being improved especially near the triple point, and valid down to 130 K sublimation temperature. It provides an absolute entropy reference value for liquid water at the triple point.