If I renounce my American citizenship can I become British citizen?

If you renounced your US citizenship you would be classified as stateless. You cannot become a Norwegian citizen until you have lived legally in Norway for a period of 7 years in any 10 year period. You could not live legally in Norway if you were considered stateless as you would have no passport to qualify for a work or residence visa to begin with.

You also need to pass a Norwegian language proficiency exam.

You cannot renounce your U.S. citizenship unless you have another one. So once you become a Norwegian citizen, Norway, being among the few countries that does not allow its citizens to hold another citizenship, you'll have to formally renounce your U.S. citizenship at the U.S. consulate in Stockholm (now handling all cases from Scandinavia).

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