(1) Background: Little is known about the effects of SARS-CoV-2 on the placenta, and whether the maternal inflammatory response is transmitted vertically. This research aims to provide information about the effects of SARS-CoV-2 infection on maternal and fetal immunity. (2) Methods: We have studied placental changes and humoral and cellular immunity in maternal and umbilical cord blood (UCB) samples from a group of pregnant women delivering after the diagnosis of SARS-CoV-2 infection during pregnancy. IgG and IgM SARS-CoV-2 antibodies, Interleukin 1b (IL1b), Interleukin 6 (IL6), and gamma-Interferon (IFN-γ), have been studied in the UCB samples. Lymphocyte subsets were studied according to CD3, CD8, CD4, CD34, and invariant natural Killer T cells (iNKT) markers. We used in situ hybridization techniques for the detection of viral RNA in placentas. (3) Results: During the study period, 79 pregnant women and their corresponding newborns were recruited. The main gestational age at the time of delivery was 39.1 weeks (SD 1.3). We did not find traces of the SARS-CoV-2 virus RNA in any of the analyzed placental samples. Detectable concentrations of IgG anti-SARS-CoV-2 antibodies, IL1b, IL6, and IFN-γ, in UCB were found in all cases, but IgM antibodies anti-ARS-CoV-2 were systematically undetectable. We found significant correlations between fetal CD3+ mononuclear cells and UCB IgG concentrations. We also found significant correlations between UCB IgG concentrations and fetal CD3+/CD4+, as well as CD3+/CD8+ T cells subsets. We also discovered that fetal CD3+/CD8+ cell counts were significantly higher in those cases with placental infarctions. (4) Conclusion: we have not verified the placental transfer of SARS-CoV-2. However, we have discovered that a significant immune response is being transmitted to the fetus in cases of SARS-CoV-2 maternal infection.