In order for me to still feel like "me" afterward, there would have to be a linking stage during which I could get used to this digital extension of my mind, like an artificial limb for me to get used to as part of my body. Then, as the biological mind withered and died, like pouring a liquid from one container to another, the digital brain would come to represent a larger and larger part of "me", finally containing all of my consciousness.

You could compare the process to how the cells of your body (in particular your brain) eventually all get replaced, yet the stream of consciousness flows uninterrupted through a number of these replacements.