Immortality of the soul just means a soul atman; the spirit was immortal from beginning of time. A soul atman was indestructible, could neither be created or destroyed.
Per se Hinduism scriptures the present journey of life in human form always belonged to soul atman within. Human form was only a medium for soul atman to cleanse itself of dross impurities within. Our soul atman was the actual master and controller of human body all the time.
Hinduism scriptures particularly Bhagavad Gita say other than God Almighty nothing existed in cosmos. And what is God Almighty, cluster of all souls’ atmans in cosmos in their pure form.
With occurrence of big bang when God Almighty, Brahman explodes self all pure souls’ atmans started their cosmic sojourn! And in their cosmic journey they gathered impurities. To cleanse self of dross impurities within started cosmic life cycle of every soul atman.
The presence of soul atman in human body can never be denied. Mounted with a bloated ego human beings failed to realize presence of soul atman within. Those indulging in spirituality, seekers finally unveiled this cosmic truth.
The purpose of spiritual journey always is emancipation from cycle of birth and death at the earliest. This becomes possible when indulging in spirituality human beings succeed in gaining enlightenment followed by moksha salvation in their lifetime.
Human beings gaining enlightenment and finally moksha salvation announces end of cosmic journey for soul atman within. The immortal liberated soul atman finally enters abode of God (Vaikuntha in Hinduism).