The poet does not really believe the bird he hears is immortal. He is using what is termed "poetic license." Poetry does not have to be literally true. This is also called a "poetic conceit." He is only pretending to think this particular bird is immortal. Here is what suggests this "poetic conceit":

Thou wast not born for death, immortal Bird!
No hungry generations tread thee down;
The voice I hear this passing night was heard
In ancient days by emperor and clown:
Perhaps the self-same song that found a path
Through the sad heart of Ruth, when, sick for home,
She stood in tears amid the alien corn;
The same that oft-times hath
Charm'd magic casements, opening on the foam
Of perilous seas, in faery lands forlorn.

The reason Keats can pretend the bird is immortal is the fact that the nightingale, like most song birds, always sings the same song. Robert Browning comments on this truth in his poem "Home-thoughts, From Abroad."

That's the wise thrush: he sings each song twice over:
Lest you should think he never could re-capture
The first fine careless rapture!

Keats was not concerned about literal reality. He enjoyed using his mind for indulgence and escape, as he tells us in a poem titled "Fancy" which begins with these lines:

Members of the nightingale's species have undoubtedly been singing the same notes for ages, perhaps ever since the species first existed thousands of years ago. That is what leads Keats to think back in time as far as he is able to travel in his imagination, drawing mainly upon his reading. This takes him back to ancient Roman times, and then further back to biblical times, and finally to times even older than the Bible, or to times that only exist in fairy tales, which often begin with the words "Once upon a time." The nightingale who is allegedly singing to the poet as he is composing his ode is thus the same nightingale who sang to Ruth in the Old Testament. But this is not literal truth; it is a poetic conceit. It is something Keats would like to believe--and he can believe it if he wants to because it is his poem and his imaginary world. He can use words to describe anything he can imagine. That, in fact, was the essence of Romanticism, wasn't it?

It is important, when answering this question, to consider how the nightingale functions symbolically in this excellent Romantic poem, rather than just literally. Of course, the speaker is not actually saying that the nightingale cannot die, rather it is saying that what the nightingale represents--what it stands for--is eternal. Let us remind ourselves of what is said in the seventh stanza:

Thou wast not born for death, immortal Bird!
No hungry generations tread thee down;
The voice I hear this passing night was heard
In ancient days by emperor and clown...

There is a clear contrast in the poem made between men and the nightingale. Men are shown to live in a world characterised by pain and suffering, the "hungry generations" that "tread" us down. However, the nightingale is shown in this poem to live in an alternative realm a realm that is free from suffering and pain. This is because in this poem the nightingale functions as a symbol of enduring art and the eternal beauty of nature. As such, the nightingale, its song and what it represents is immortal compared to the brief lives of men.