So, how about, you get a small computer to take over the job of a set of neurons that are about to kick the bucket. You hook up your computer to all the little synapses that the old cells were talking to and you start the simulation. Can you tell the difference?
If not, repeat. Over many iterations, your brain should be completely replaced with computers, all without you even noticing so there is a continuity of consciousness. Now you are immortal (until the power cuts out). Your bits can be replaced indefinately.
Erm, 'you' wouldn't experience it.
A different entity would take over, and you wouldn't know.