I think immortality is too vague a term to use in any non-religious argument.
The mythical idea of immortality (Greek for example) seems to imply not only never dying, but never aging (but able to grow and change), never getting tired (but able to sleep and dream), never being hungry or thirsty (but still having an appetite), and so on.
In other words, living forever with none of the disadvantages.
In current religions, it's already taken care of. The soul is beyond the mortality of the flesh, God will take care of it, Heaven is wonderful, etc.
But all of this is mythical.
I would settle for not aging, and having the freedom to end my life if I see no point in continuing. Being able to do some kind of reset would also be nice.
The only non-religious concept of immortality that approaches mythical immortality is part of the Singularity: mind uploading, personality editing and the like.
The SF author Norman Spinrad wrote a novel "Deus X", in which a form of mind uploading is possible.
Note: by form, I am referring to the arguments as to whether an "uploaded" mind is actually me (a continuation of my consciousness as complete as the one that happens every day) or just "someone" who remembers being me.
Which case applies depends (speculating wildly) on the mechanics of the upload. In Spinrad's novel, uploading involves examining a living brain (non-destructively) and creating a program that emulates the original person. This would come under the "remembers" case, since the original is still around after the process is complete.
In the novel, the Catholic Church is trying to decide if uploads have souls. They decide to upload a dying priest who rejects this idea and have him argue as an upload that he has no soul.
This is done, and interesting things ensue.