  The Continuous-Variable (CV) quantum state of light allows for the
teleportation of a Discrete-Variable (DV) photonic qubit. Such an operation is
useful in the realm of hybrid quantum networks. However, it is known that the
teleportation of a DV qubit via a Gaussian CV resource channel is severely
limited under channel loss, with a teleportation fidelity beyond the classical
limit unattainable for losses exceeding a small threshold of 0.5 dB. In this
work, we present a new non-deterministic teleportation protocol that combines a
Gaussian CV resource channel with a modified form of the Bell State Measurement
that accommodates a DV-qubit fidelity beyond the classical limit for channel
losses up to 20~dB. Beyond this orders of magnitude improvement, we also show
how the use of non-Gaussian operations on the CV resource channel can lead to a
DV-qubit fidelity approaching unity for any channel loss.
