We solve the Klein-Gordon equation with a new anharmonic oscillator potential and present the exact solutions.It is shown that under the condition of equal scalar and vector potentials, the Klein-Gordon equation could be separated into an angular equation and a radial equation. The angular solutions are the associated-Legendre polynomial and the radial solutions are expressed in terms of the confluent hypergeometric functions. Finally,the energy equation is obtained from the boundary condition satisfied by the radial wavefunctions.