Zaremba was born in Poznań, Poland, son to Oskar Bielawski and Elżbieta Immerdauer.

He started writing for the newspaper Dagens Nyheter in 1989 and in 1994 he was recruited to the editorial team of the now defunct magazine Moderna tider.

In 1997, Zaremba became famous after he wrote a series of articles in Dagens Nyheter about the compulsory sterilization program active in Sweden from the mid-1930s until the mid-1970s.

In 2005 he wrote a series of articles about a labor dispute involving foreign construction workers in Vaxholm.

In 2019, he was the recipient of the Ryszard Kapuściński Award for Huset med de två tornen (A House with Two Turrets).