Between October 2013 and April 2014, more than 30,000 cases of Zika virus (ZIKV) disease were reported in French Polynesia. ZIKV has also been reported in Africa and Asia, and in 2015 the virus spread to South America and the Caribbean. Infection with ZIKV has been associated with neurological complications including Guillain-Barre Syndrome (GBS) and microcephaly, which led the World Health Organization declare a Public Health Emergency of International Concern in February 2015. To better understand the transmission dynamics of ZIKV, we used a mathematical model to examine the 2013-14 outbreak on the six major archipelagos of French Polynesia. Our median estimates for the basic reproduction number ranged from 1.9-3.1, with an estimated 11.2% (95% CI: 10.1-12.9%) of total infections reported. As a result, we estimated that 86% (95% CI: 75-93%) of the total population of the six archipelagos were infected during the outbreak. There were 42 GBS cases reported during the ZIKV outbreak in French Polynesia, but the presence of a large number of unreported ZIKV infections could have implications for the design of case-control studies to further investigate a possible association between the two. Based on the demography of French Polynesia, our results also imply that if ZIKV infection provides complete protection against future infection, it would take 15-20 years before there are a sufficient number of susceptible individuals for ZIKV to re-emerge, which is on the same timescale as the circulation of dengue virus serotypes in the region. Our analysis suggests that ZIKV may exhibit similar dynamics to dengue virus in island populations, with transmission characterised by large, sporadic outbreaks with a high proportion of asymptomatic or unreported cases.