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Saint-Maximin, a French Under-21 international, moved to the Premier League club on a six-year contract for a fee reported to be £20 million.
The 22-year-old made his professional debut for Saint-Etienne at the age of 16 and has gone on to make over 100 Ligue 1 appearances.
He has played for Monaco, Bastia and Nice, while also going on loan to Hannover.
"I'm delighted to bring Allan to St James' Park. He is a very talented young player with a big future ahead of him," Newcastle manager Steve Bruce told the club's website.
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"He has all the attributes you would want in an attacking player, including lightning pace, so I'm sure he'll excite everybody. He is a great addition to Newcastle United and the Premier League."
Newcastle have the option of signing Willems on a permanent basis at the end of the upcoming season, which begins next weekend.
"Jetro brings a wealth of experience into the dressing room and I'm very pleased to have brought him here," Bruce said.
"He has played at the highest level and is naturally left-footed so he'll give us another strong option on that side."
The 25-year-old left-back has played 22 times for the Netherlands and arrives at St James' Park ahead of their opening league match against Arsenal next Sunday.
Willems started his career with Sparta Rotterdam before in 2011 joining PSV Eindhoven, where he won two Dutch Eredivisie titles.
He moved to Frankfurt in 2017 and won the German Cup that season.