Many studies have shown that neonates of smoking mothers have a lower birth weight, but several issues remain poorly studied, e.g., the effects of giving up smoking or the combined effects of smoking and maternal obesity. Therefore, we evaluated a prospective cohort of 912 mothers in a single pregnancy, recruited in Poland, in 2015−2016. In the cohort, we recorded 72 (7.9%) newborns with birth weight <10th percentile, 21 (2.3%) fetal growth restriction (FGR) cases, and 60 (6.6%) low birth weight (LBW, <2500 g) newborns. In the cohort, 168 (18.4%) women smoked before pregnancy; the mean number of cigarettes/day was 10.8 (1–30), and the mean number of years of cigarette smoking was 8.5 (1–25). Among smokers, 57 (6.3%) women smoked in the first trimester. Adjusted odds ratio (AOR) of newborn outcomes (with 95% confidence intervals, CI) was calculated in multi-dimensional logistic regressions. Compared to participants who had never smoked, smoking before pregnancy was associated with a higher odds ratio of birth weight <10th percentile (AOR = 1.93, CI: 1.08–3.44, p = 0.027), but the result for LBW (AOR = 2.76, CI: 1.05–7.26, p = 0.039) and FGR (AOR = 1.13, CI: 0.38–3.36, p = 0.822) had the wider confidence interval or was insignificant. Effects of smoking cessation before pregnancy were statistically insignificant for the studied outcomes. Smoking in the first trimester was associated with a higher risk of birth weight <10th percentile (AOR = 4.68, CI: 2.28–9.62, p < 0.001), LBW (AOR = 6.42, CI: 1.84–22.36, p = 0.004), and FGR (AOR = 3.60, CI: 0.96–13.49, p = 0.057). Smoking cessation in the second/third trimester was associated with a higher odds ratio of birth weight <10th percentile (AOR = 4.54, CI: 1.58–13.02, p = 0.005), FGR (AOR = 3.36, CI: 0.6–18.74, p = 0.167), and LBW (AOR = 2.14, CI: 0.62–7.36), p = 0.229), to a similar degree to smoking in the first trimester. The odds ratios were higher in the subgroup of pre-pregnancy body mass index ≥25 kg/m2 for the risk of birth weight <10th percentile (AOR = 6.39, CI: 2.01–20.34, p = 0.002) and FGR (AOR = 6.25, CI: 0.86–45.59, p = 0.071). The length of cigarette smoking time was also the risk factor for studied outcomes. Conclusions: Smoking in the first trimester increased the studied risks, and the coexistence of excessive maternal weight increased the effects. Smoking cessation during the second/third trimester did not have a protective effect.