Then he'll be fine in Sweden, the Swede's won't extradite anyone who will face the death penalty.

See here: http://www.sweden.gov.se/content/1/c6/01/54/35/77809ec6.pdf. Page 3, Section 12, Point 3.

Worth noting that this is the text of the "THE EXTRADITION FOR CRIMINAL OFFENCES ACT (1957:668)", rather than some statement of government position and therefore is binding in Swedish law. The government simply couldn't send him to the US without suitable assurances.