Woltemade, who signed a "long-term" contract at St James' Park, only joined Stuttgart from Werder Bremen last summer, and despite not being a regular starter scored 17 goals in 33 games, including the opening goal in their German Cup final triumph.
The imposing 6ft 6in striker lit up the Under-21 Euros last month and ended as the tournament's top scorer with six goals. He also scored in the final as Germany lost 3-2 to England.
Newcastle have agreed to pay £64.8m (€75m) plus £4.3m (€5m) in add-ons.
Woltemade, who will wear the number 27 shirt, told the club's website: "I'm really happy to be at this amazing club. From the first contact, I felt like the club really wanted me and had big plans for me.
"It's a big step in my life to leave Germany but everybody has welcomed me so well and it already feels like family. I have a really good feeling from speaking to the head coach that this is the right place for me to find my best level.
"I know the stadium from watching games on television - it looks amazing and I know the atmosphere is crazy. I'm really excited to play and start scoring goals here."
Newcastle manager Eddie Howe, who has had a frustrating time in the transfer market, said Woltemade, was an ideal fit.
"We're delighted to get Nick's signing over the line so quickly. He fits the profile for exactly what we have been looking to add to our attacking options," Howe said.
"He's strong in a lot of areas - he has great technical ability and has proven himself to be a real threat in one of Europe's top leagues - but he's also still at an age where he has plenty of room to develop and grow here.
"Nick is a great character too and we're really pleased to welcome him to the group."
The transfer could encourage Liverpool to make an increased offer for Newcastle's wantaway striker Alexander Isak after seeing a £110m bid rejected earlier this month.
Isak, whose goals fired Newcastle to Champions League qualification and League Cup victory last season, has been training on his own since making clear his desire to move to Anfield.
Howe had hinted for the first time on Friday he might reluctantly be open to allowing Isak to leave, because of the impact a sale would have on Newcastle's profit and sustainability calculations.
Newcastle were forced to sell Elliot Anderson and Yankuba Minteh last year in order to avoid a points deduction, and a sizeable return on their investment in Isak would ease any Profit and Sustainability Regulations (PSR) concerns.