Bett signed for Fylkir in November 2009, agreeing on a 2-year contract, moving on to Reynir S. after those 2 years.
He spent the majority of his playing career with Fylkir, making more than 150 league appearances for the club.
Ásgeirsson was born in Reykjavík, and played for Fylkir's youth teams, before joining lower leagues team UMF Selfoss on loan for two consecutive years in 2007 and 2008.