[3] He began his career with ÍA in his native Iceland before a three-year spell at Norwegian side Lillestrøm, during which time he made his international debut.

[6] After making his club debut on 22 March 2009 against Viking,[7] Björn's first season in the Tippeliga was troubled by numerous lower back injuries.

On 18 June 2012, Björn's club officially announced that they had accepted an undisclosed bid for him from English side Wolverhampton Wanderers, believed to be in the region of £2.4 million.

[12] The Icelandic striker made his Wolves debut as a substitute in an opening day defeat at Leeds United.

On 1 January 2016, he came on as a 67th-minute substitute for James Henry in a 1–0 Championship win at Brighton & Hove Albion, his first Wolves appearance for over two years.

[16] Björn went on to often be used during the second half of the season, frequently as a lone forward, but failed to score during these 15 appearances, meaning his final goal for Wolves had come almost three years prior in August 2013.

[22] Björn has represented Iceland at numerous youth levels, most recently playing for their Under-21 team in the qualifying competition for the 2013 European Under-21 Championship.

He has seven caps for the full Icelandic national team, having made his debut as a substitute in a 1–0 win over Cyprus during Euro 2012 qualification on 6 September 2011.