  The efficiency of a Bell-state measurement on photon pairs is bound to 50\,\%
due to the number of Bell states that can be distinguished using linear optics.
Here we present the implementation of a protocol that allows us to distinguish
all four Bell states by the use of a single-ion quantum memory and heralded
absorption as state-selective measurement. The protocol is implemented in two
steps. First we demonstrate the state-preserving mapping of a photonic qubit
onto the quantum memory, verified by the preservation of entanglement in the
process. Then we demonstrate the full Bell state projection between a memory
qubit and an incoming photonic qubit, by applying it for atom-to-photon quantum
state teleportation.
