To overcome these limitations, we develop a quantitative cervical elastography system by adding a stress sensor to a clinically used transvaginal ultrasound imaging system. In an imaging session, we use the ultrasound system to record the cervical deformation in B-mode images and use the stress sensor to record the probe-surface stress simultaneously. We develop a feature-tracking algorithm to quantify the deformation automatically and calculate the strain. Then we estimate the cervical Young's modulus through stress-strain linear regression. In phantom experiments, we demonstrate the elastography system's high accuracy (alignment with the quasi-static compression method, p-value = 0.369 > 0.05), robustness (alignment between 60°- and 90°-contact measurements, p-value = 0.638 > 0.05), repeatability (consistency of single sonographers' measurements, coefficient of variation < 0.06), and reproducibility (alignment between two sonographers' measurements, Pearson correlation coefficient = 0.981). Applying it to pregnant participants, we observe significant softening of the cervix during pregnancy (p-value < 0.001) with the cervical Young's modulus decreasing 3.95% per week. We estimate that geometric mean values of cervical Young's moduli during the first (11 to 13 weeks), second, and third trimesters are 13.07 kPa, 7.59 kPa, and 4.40 kPa, respectively. The proposed system is accurate, robust, and safe, and enables longitudinal measurements and comparisons between examiners. The system applies to different ultrasound machines with minor software updates, which allows for studies of cervical softening patterns in pregnancy for larger populations, facilitating insights into conditions such as preterm birth.