"Kelly has done an outstanding job representing our Nation and I have no doubt that, under her leadership, our Country will be represented at the highest level," Trump tweeted. "Congratulations to Kelly and her entire family!"
If confirmed for the role, Craft will succeed Nikki Haley, who abruptly announced her resignation in October last year.
Haley offered Craft her congratulations on Twitter following the announcement. "She’s done a great job representing us as @USAmbCanada and we know she’ll be a strong voice for America at the United Nations," Haley tweeted.
“I am grateful to President Trump and Secretary [Mike] Pompeo for the trust they placed in me for considering me for the position of U.S. ambassador to the United Nations," Nauert said in the statement released by the State Department on Saturday. "However, the past two months have been grueling for my family and therefore it is in the best interest of my family that I withdraw my name from consideration."
A State Department source told Fox News at the time that the confirmation process -- on top of traveling around the world and between Washington and New York to see family -- grew to be too much for Nauert.
Craft was sworn in as the ambassador to Canada in September 2017 after being nominated to the position by Trump, according to her biography on the Embassy's website.
"Ambassador Kelly Craft is a leader, entrepreneur, and philanthropist who has made community service and improving education the cornerstone of her career," her biography said.
The website lists several positions that Craft has held in Kentucky, including leadership roles with the Salvation Army of Lexington. She has also served on the University of Kentucky Board of Trustees, and started a marketing and business advisory company, it said.
"In 2007, President George W. Bush appointed Ambassador Craft as an alternate delegate to the United Nations General Assembly," according to her biography.
Pompeo issued a statement saying he was "very pleased that President Trump has announced his intent to nominate Ambassador Kelly Craft to be our next U.S. ambassador to the United Nations.
"Ambassador Craft has been an outstanding advocate for America’s national security and economic interests in Canada and she is extremely well-qualified to do the same at the United Nations," he continued. "I look forward to her confirmation and continuing to work with her at the United Nations."
Senate Majority Leader Mitch McConnell, R-Ky., hailed the president's nomination, labeling Craft "an exceptional choice for this critical post."
"Kelly Craft is a strong advocate for American interests and will be a powerful representative of our great nation at the U.N.," his statement read. "She has a long record of service to her state and the nation and I'm confident she will continue to serve with distinction as America's voice to the world at the United Nations. I was proud to recommend this remarkable Kentuckian to President Trump."
McConnell's statement noted that Craft "is a native of Glasgow, Kentucky."
Fox News' Mike Emanuel, Nicholas Kalman and Rich Edson contributed to this report.