This is the part where things get rather difficult to theorize. If the transfer of mind and memory to an artificial construct were possible, would there then be two separate versions of yourself with the same consciousness? Or would you be conscious of your original body only while the duplicate believes itself to also be the original?

Assuming that at the time of the Singularity, we've already created a Matryoshka Brain, a particular Deus Est Machina of sorts. >! (A literal God AI, Deus Est Machina roughly translates to God is Machine. A Matryoshka Brain is a supercomputer-powered AI inside a Dyson Sphere powered by the consumption of an entire star that would theoretically be able to hold conversation with the entire world without using 0.1% of its processing power.) !< As well as assuming that our consciousness is tied to our own minds and memories, then death becomes a minor inconvenience while a new artificial you is created after whatever mishap you endured occurs. Human evolution is then flung forwards as time suddenly becomes endless.

The Ship of Theseus method, then?

Any sufficiently advanced technology is indistinguishable from Magic. Give it another few decades or so as technology marches onward so does the Singularity and humanity’s technological evolution. When one become the same evolution is only limited by the limitations of the technology that binds it.