The 16-year-old, who stands at 1.94m, arrives from Stjarnan Gardabaer in Iceland and will initially join Dortmund's under-19 side.
The move had been effectively agreed for some time, with the final holdup being FIFA clearance for the international transfer of a minor, which has now been granted. Only a training compensation fee changes hands as part of the deal, in line with regulations covering moves of this kind.
Gardarsson made his senior debut for Stjarnan in May and has since gone on to make three appearances in Iceland's top flight, form that has helped mark him out as one of the country's most promising talents in his age group. He has also become a regular for Iceland's under-17 side, with eight caps and two goals to his name at that level so far.
Dortmund's youth academy director Thomas Broich welcomed the signing, praising the physical profile the young striker brings to Brackel.
"Bjarki is a striker who is physically present and purposeful, and who brings a good sense for the penalty box," Broich said.
"We've had him on our radar for a while now, and we're delighted that he has chosen to join Borussia Dortmund."
While Gardarsson will begin life in Germany with the under-19s, the club's long-term hope is that his combination of physical presence, two-footedness and composure in front of goal can eventually translate into a path towards the first team, continuing a recruitment strategy that has increasingly focused on identifying and developing international teenage talent before it attracts wider attention.
The move continues a busy summer for Dortmund's academy, which has now brought in three internationally capped teenagers in quick succession. Yamamoto, 18, and Kante, 16, both arrived in recent weeks as part of the same push to strengthen the club's underage sides with talent recruited from beyond Germany's borders.
Gardarsson will link up with under-19 coach Felix Hirschnagl's squad immediately, with the aim of building match sharpness before any decision is made on his path towards senior football at Signal Iduna Park.