The 20-year-old has signed a four-year contract and will join up with Celtic on January 1.
Fridjonsson impressed Celtic boss Neil Lennon during trials at the club.
"I'm really delighted to sign for such a big club like Celtic," Fridjonsson said. "This is just a dream come true for me.
"It is unbelievable to be here and I can't wait to get started. I was buzzing when I heard of their interest and when Celtic came in for me there was nothing for me to decide. It was just Celtic.
"And playing for a big club like Celtic will certainly increase my chances of playing for the national side and hopefully I can break into the squad ahead of the Euro 2016 qualifiers."
Fram chairman Sverrir Einarsson said: "This is a very good piece of business for Celtic and it is very good business for us."