We present a novel Bell-state analyzer (BSA) for time-bin qubits allowing the detection of three out of four Bell states with linear optics, two detectors, and no auxiliary photons. The theoretical success rate of this scheme is 50%. Our new BSA demonstrates the power of generalized quantum measurements, known as positive operator valued measurements. A teleportation experiment was performed to demonstrate its functionality. We also present a teleportation experiment with a fidelity larger than the cloning limit.