Can a company hire you before doing a background check?

I have been looking for a job for the past 5 months and finally a company called and ask that I start in a week. They gave me the hours that I will work and the pay then said, contingent on a background check. WELL HELL, I waited a week to hear from them and now they call me said all this and now want to do a background check -- they should have done that before asking me to work for them. Im upset because I have 1 little incidence that happened 19 years ago and it still comes up -- but on the application I put down (never been arrested). My question is do you think a company can hire you before doing a background check?