Is immortality with limited capacity for memory really immortality?

It is currently believed that the human brain has unimaginably vast capacity for memory (the number of possible connections in the human brain outnumbers the stars in the entire Universe), but it is not infinite. That means, when a way is discovered to make the human body immortal, our older memories will fade as we gain new ones.

Assume that you are made immortal, and that you have the capacity to retain, say, the last 100 years or so of your experiences. That means, 100 years from now, you will remember absolutely nothing of what you remember today. There is no single point where you will have changed into an entirely new person, but a snapshot of your memories taken between any two points 100 years apart will contain absolutely nothing which is the same.

Do you think you would be the same person? There would be a constant continuity from day to day, but hundred-year old memories would simply fade silently away without your awareness, gradually changing your identity. For those of you who believe in a spirit, would you have the same spirit throughout the thousands of years of your life even though you would change utterly again and again? And lastly, if such immortality becomes possible – and it may, within your lifetime – would you choose to accept it?

(For the philosophically minded, this is a special case of the Ship of Theseus paradox.)