Objective: Strain elastography and shear wave elastography are two commonly used methods to quantify cervical elasticity; however, they have limitations. Strain elastography is effective in showing tissue elasticity distribution in a single image, but the absence of stress information causes difficulty in comparing the results acquired from different imaging sessions. Shear wave elastography is effective in measuring shear wave speed (an intrinsic tissue property correlated with elasticity) in relatively homogeneous tissue, such as in the liver. However, for inhomogeneous tissue in the cervix, the shear wave speed measurement is less robust. To overcome these limitations, we develop a quantitative cervical elastography system by adding a stress sensor to an ultrasound imaging system. Methods: In an imaging session for quantitative cervical elastography, we use the transvaginal ultrasound imaging system to record B-mode images of the cervix showing its deformation and use the stress sensor to record the probe-surface stress simultaneously. We develop a correlation-based automatic feature tracking algorithm to quantify the deformation, from which the strain is quantified. After each imaging session, we calibrate the stress sensor and transform its measurement to true stress. Applying a linear regression to the stress and strain, we obtain an approximation of the cervical Young's modulus. Results: We validate the accuracy and robustness of this elastography system using phantom experiments. Applying this system 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.