A scheme for teleporting an unknown single-qubit photonic state in a nearly deterministic and controlled manner is proposed via the weak cross-Kerr nonlinearities. The functions of the complicated CNOT operations and Bell-state analysis can be realized easily with the help of quantum nondemolition measurement technique. This improvement makes the present scheme more efficient than the schemes using nonunitary projective measurements. Discussions about this scheme show that it is feasible with the current experimental technology in optics.