The intermolecular potential for the van der Waals complex of the carbon monoxide cation with helium is studied by means of the partially spin adapted coupled cluster RCCSD(T) method and the aug-cc-pVXZ family of basis sets. In the ground electronic state, correlated with the lowest electronic asymptote $X {^2\Sigma^+}$ of the monomer CO$^+$, the complex He($^1S$)-CO$^+$($^2\Sigma^+$) has a nonlinear equilibrium structure with a Jacobi angle of about 46 deg and a binding energy of about 275 cm$^{-1}$. For the complex He($^1S$)-CO$^+$($A {^2\Pi}$) we find equilibrium Jacobi angles of 78 deg and 90 deg and electronic binding energies of about 160 cm$^{-1}$ and 303 cm$^{-1}$ for the $A^{\prime\prime}$ and $A^{\prime}$ components, respectively, coalescing into the $\Pi$ state at linearity. Two-dimensional intermolecular potential energy surfaces are constructed for the ground electronic state and used to compute rotation-vibration states up to J=10 with the numerically exact discrete variable representation (DVR) technique. The He-CO$^+$ complex is found to have 19 bound even-parity J=0 states and 16 bound odd-parity J=1 states and to exhibit strong angular-radial coupling and quasilinear behaviour.