Human infections by Zika virus (ZIKV), a mosquito-borne flavivirus, are associated with a current widespread outbreak in the Americas, and have been associated with neurological complications and adverse fetal outcomes such as microcephaly in pregnant women. A suitable non-human primate model is urgently needed. To evaluate ZIKV infectivity, pathogenesis, and persistence, we inoculated 4 marmosets with ZIKV and followed them by clinical monitoring and serial sampling of body fluids for up to 11 weeks. We found that marmosets experimentally infected with ZIKV reproduced key features of the human disease, including (1) asymptomatic infection, (2) brief period of detectable virus in serum (<1 week), (3) detection in other body fluids (urine, saliva, semen, and stool) for at least 2 weeks following acute infection, and (4) persistence in lymph nodes, but not other tissues, at 1 month post-infection. ZIKV-positive saliva and serum samples, but not urine, were found to be infectious in cell culture. By day 6 post-inoculation, most marmosets exhibited detectable neutralizing antibody responses concurrent with activation of NK cell and B cell subsets and an increase in circulating cytokines associated with type II interferon signaling, Transcriptome profiling revealed enrichment of immune responses to active viral infection, with up-regulation of both type I and II interferon signaling pathways, and uncovered potential host biomarkers. These results suggest that a New World monkey model of acute ZIKV infection mimics the human disease, and is likely to be useful for testing of drug and vaccine candidates.