Background/Objectives: Zika disease is caused by the Zika virus (ZIKV) and represents a major public health problem because of the complications in newborn babies from mothers who were infected during pregnancy. It is estimated that 80% of infected pregnant women are asymptomatic, which complicates the identification of infected individuals. In this study, we aimed to detect ZIKV in asymptomatic pregnant women and the effects in the newborns were analyzed. Methods: The presence of ZIKV was evaluated through endpoint reverse transcription–polymerase chain reaction (RT-PCR) in 114 blood samples from pregnant women treated at two hospitals in the state of Veracruz, Mexico. There was a follow-up of the participants until the birth of their newborns. Results: ZIKV RNA was detected in 4.4% (n = 5) of cases. In two positive cases, two consecutive samples were obtained, and one case of persistence of ZIKV in serum after 90 days after delivery was identified. A total of 80% of the positive cases were identified after the third trimester of pregnancy and 20% after the second trimester. Although ZIKV was shown to be a risk factor for low weight and low size at birth and prematurity, after adjustment for other variables, it did not show a significant association. In contrast, preeclampsia/eclampsia was identified as a significant risk factor for low birth weight. Conclusions: The prevalence of ZIKV found in this study suggests a latent circulation of this virus and highlights the importance of epidemiological surveillance in endemic zones. The prolonged viremia that was found suggests the need for more research because of the high impact which can mean the possible dissemination of the virus to the vector.