And now we get to the heart of the matter, which is the most elegant and subtle aspect of the Buddha's Dhamma. Simply stated: when ignorance ceases, belief in self simultaneously ceases. And when there is no self to be found, then there is no self to die or take birth. This right here is 'deathless.' And it is precisely this that the Buddha is declaring when he says to Mogharaja in the Sutta Nipata:
View the world, Mogharaja, as empty —
always mindful to have removed any view about self.
This way one is above & beyond death.
This is how one views the world so as not to be seen by Death's king.
It is the notion of a 'self' as either possessing or within 'name and form' (nama-rupa) which results in death. When one abandons the belief in a self — any and all notions of 'I am' — then there is no self to die. This 'stilling of the currents of construing' over one's imagined self, and the resulting peace that is empty of birth, aging, and death, is straightforwardly presented in MN 140: Dhatu-vibhanga Sutta:
'He has been stilled where the currents of construing do not flow. And when the currents of construing do not flow, he is said to be a sage at peace.' Thus was it said. With reference to what was it said? 'I am' is a construing. 'I am this' is a construing. 'I shall be' is a construing. 'I shall not be'... 'I shall be possessed of form'... 'I shall not be possessed of form'... 'I shall be percipient'... 'I shall not be percipient'... 'I shall be neither percipient nor non-percipient' is a construing. Construing is a disease, construing is a cancer, construing is an arrow. By going beyond all construing, he is said to be a sage at peace.
Furthermore, a sage at peace is not born, does not age, does not die, is unagitated, and is free from longing. He has nothing whereby he would be born. Not being born, will he age? Not aging, will he die? Not dying, will he be agitated? Not being agitated, for what will he long? It was in reference to this that it was said, 'He has been stilled where the currents of construing do not flow. And when the currents of construing do not flow, he is said to be a sage at peace.'
Truly, 'a sage at peace is not born, does not age, does not die.' In this way, when ignorance ceases, the entire complex of dependent origination bound up with suffering also ceases. Ud 1.3: Bodhi Sutta:
Now from the remainderless fading and cessation of that very ignorance comes the cessation of fabrications.
From the cessation of fabrications comes the cessation of consciousness.
From the cessation of consciousness comes the cessation of name-and-form.
From the cessation of name-and-form comes the cessation of the six sense media.
From the cessation of the six sense media comes the cessation of contact.
From the cessation of contact comes the cessation of feeling.
From the cessation of feeling comes the cessation of craving.
From the cessation of craving comes the cessation of clinging/sustenance.
From the cessation of clinging/sustenance comes the cessation of becoming.
From the cessation of becoming comes the cessation of birth.
From the cessation of birth, then old age and death, sorrow, lamentation, pain, distress, and despair all cease.
Such is the cessation of this entire mass of stress and suffering.
And this then, is the meaning of 'unconditioned:' when self and phenomena (i.e. the 'I' and 'mine') are abandoned through the fully integrated threefold practice of ethical conduct, meditation, and discernment, just this is dispassion (viraga). Just this is 'cessation' (nirodha). Just this is the ending of the effluents (asava). Just this is unborn (ajata), unoriginated (abhuta), unmade (akata), uncreated (asankhara), endless (ananta), indestructible (apalokita), permanent (dhuva), and yes, deathless (amata). The notion of the birth and death of the self was never anything more than ignorant fantasy. Just this is the timeless (akalika) Dhamma.