Thank you for reaching out to Constant Contact API Support.
At this time there is not a way to make a direct copy of an email using the Constant Contact API.
If you are trying to make a copy of a custom XHTML email, you can use the API to GET the email content of the campaign (as shown here: http://developer.constantcontact.com/docs/email-campaigns/email-campaign-resource.html), and can then POST a new email campaign using that HTML code (as shown here: http://developer.constantcontact.com/docs/email-campaigns/email-campaigns-collection.html?method=POS...).
Please note that this will not work with emails created using our email wizard editor, as our API will not provide the HTML code for emails created this way. Also, these emails would not work an any case as they contain code that is not supported by our custom email template type.
If you have any other questions, please let me know!
API Technical Support Specialist