The 81-year-old Democrat posted on X that he would remain in his role as president and commander-in-chief until his term ends in January 2025.
He endorsed Vice President Kamala Harris to replace him as his party's candidate in the upcoming November election.
Early on Monday Mr Albanese posted a message on X thanking Mr Biden and sharing the president's tweet.
"Thank you for your leadership and ongoing service President Biden," Mr Albanese wrote.
"The Australia-US Alliance has never been stronger with our shared commitment to democratic values, international security, economic prosperity and climate action for this and future generations."
Thank you for your leadership and ongoing service President Biden. The Australia-US Alliance has never been stronger with our shared commitment to democratic values, international security, economic prosperity and climate action for this and future generations. — Anthony Albanese (@AlboMP) https://t.co/gkAggLguAjJuly 21, 2024
Mr Biden wrote that it had been the greatest honour of his life to serve as US president.
"And while it has been my intention to seek re-election, I believe it is in the best interest of my party and the country for me to stand down and to focus solely on fulfilling my duties as President for the remainder of my term."
Mr Biden's announcement follows public and private pressure from Democratic lawmakers and party officials to quit the race after his poor performance in a June 27 televised debate against 78-year-old Republican rival Donald Trump.
with Reuters
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