We investigate a type of one-dimensional quasi-exactly solvable double-well potential whose analytical solution can be constructed in terms of the Heun functions. It is shown that for certain special values of the potential parameters, two energy eigenvalues and eigenstates of the lowest part of the energy spectrum can be found exactly in explicit form. In addition, the Wronskian method has been applied to derive the conditions for the energy eigenvalues of the bound states. Our analytical results may find applications in the tunnelling problem for the double-well potentials.