Biden wrote to Ardern that he looked forward to "strengthening the US-New Zealand relationship" and cooperating to "overcome the greatest challenges of our time".
He added he had "fond memories" of his trip to New Zealand in 2016 "when New Zealand's legendary reputation for friendliness and hospitality was on full display".
"I wish you and all New Zealanders well on this Waitangi Day."
The letter is then signed: "Sincerely Joseph R Biden".
When Biden visited here in 2016 he was American vice-president to Barack Obama.
During the trip he had a formal meeting with then New Zealand Prime Minister John Key.
Following the meeting, Biden told reporters: "But we don't always agree. But what we do is we share the same basic values of promoting freedom, equality, and opportunity.
"And it's that kinship of spirit that drives our nations' desire – and capacity – to work together to make the world a better place. And presumptuous of me to say but I think that's exactly what we're doing together.
"It's been a productive visit, an important opportunity to celebrate what unites all of our people. And I believe that our nations are only going to grow closer as we continue to work together in the 21st century."
On Thursday, Foreign Affairs Minister Nanaia Mahuta name-checked the newly elected US President in her first major speech in the job, citing New Zealand's strong relationship with the United States.
He was the only world leader Mahuta mentioned by name in the speech she made to diplomats.
She praised some of the early moves Biden has made, including rejoining the Paris Agreement on climate change and renewing the US commitment to the World Health Organisation.
"New Zealand welcomes the recent announcements from President Biden that the United States will take a science-based approach to combating the challenge of Covid-19," Mahuta said.
She added that New Zealand's relationship with the US is an "integral defence and security partner".
"New Zealand's relationship with the United States will continue to strengthen."
Ardern also spoke warmly of Biden after his election win over Donald Trump.
"I look forward to building a close relationship with President Biden and working with him on issues that matter to both our countries," Ardern said in a statement.
"New Zealand and the United States have shared interests in addressing global challenges including climate change, the Covid-19 economic recovery, and the security, prosperity and sustainability of the Indo-Pacific and Pacific Island regions.
"We have a common investment in the international rules-based order and I welcome President Biden's intentions for the US to rejoin the Paris Agreement and halt its withdrawal from the World Health Organisation.
"President Biden is a good friend of New Zealand, and we have fond memories of his 2016 visit to our shores. Under his presidency the strong and close relationship that New Zealand enjoys with the United States will continue to develop and flourish.
"President Biden's message of unity as he takes office is one that resonates with New Zealanders and, on behalf of New Zealand, I wish him well as he begins his presidency."
