We report the experimental demonstration of a quantum teleportation protocol with a semiconductor single photon source. Two qubits, a target and an ancilla, each defined by a single photon occupying two optical modes (dual-rail qubit), were generated independently by the single photon source. Upon measurement of two modes from different qubits and postselection, the state of the two remaining modes was found to reproduce the state of the target qubit. In particular, the coherence between the target qubit modes was transferred to the output modes to a large extent. The observed fidelity is 80%, in agreement with the residual distinguishability between consecutive photons from the source. An improved version of this teleportation scheme using more ancillas is the building block of the recent Knill, Laflamme, and Milburn proposal for efficient linear optics quantum computation.