Kamiquasi on CAB-10647 - Ribbon Cable - 10 wire (15ft)

While you’d have to check for your specific use (length, voltage drop, what temperature you’re comfortable with, etc.), the datasheet specifies the strands as 28 AWG and our handy dandy little workbook has values a fair bit over 1A.
Online resources note several different values, but also all above 1A. Looks like a few decades ago 28 AWG ribbon cables were especially tested and they concluded 3A – I’d err on the cautious side and not try that, though; http://www.osti.gov/bridge/product.biblio.jsp?osti_id=6970997