The 18-year old midfielder has signed a three-and-a-half-year deal with the Swans, joining for an undisclosed fee.
King, who successfully underwent a medical on Monday, was thrilled to be making the move to the Premier League after coming through the Hearts youth academy.
"It feels brilliant to be here," the Scotland U19 international told the club's official website.
"I've left a passionate club and I'm arriving at another passionate club.
"It's always going to be hard to leave the club that you love, your home and your family as well, but coming to Swansea, where they're building something special here, it's a very good club to be at.
"The way Swansea play football is brilliant and will suit me down to a tee. I like to get the ball down and play, not force passes, and get around the pitch.
"I'm excited and I'm looking forward to my time here."
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