Woltemade 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 (1.98m) striker lit up the Under-21 Euros last month and ended as the tournament's top scorer with six goals, including in the final as England beat Germany 3-2.
Newcastle have agreed to pay £64.8m (€75m) plus £4.3m (€5m) in add-ons.
His signing could encourage Liverpool to make an increased bid for Newcastle's wantaway striker Alexander Isak after seeing an offer of £110m rejected earlier this month.
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 United head coach Eddie Howe added: "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.
"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."
His signing could encourage Liverpool to make an increased bid for Newcastle's wantaway striker Alexander Isak after seeing an offer of £110m rejected earlier this month.