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Accoding to the US State Department webpage on Jamaica, travelers entering Jamaica need to show a return ticket and sufficient funds for their visit. Visitors are not required to have visas for general vacationing, but those planning to work there or have an extended stay need to present a current US passport and visa from the Jamaican Embassy or Consulate.
The US also issues passport cards. Passport cards look like a credit card, but allow travelers by any means (except air) to enter and exit Canada, Mexico, the Caribbean and Bermuda. More information is available here.