In December 2017, a dual-polarized (DP) phased-array weather radar (PAWR) was deployed to observe precipitation in Tokyo, Japan. The DP-PAWR has the following characteristics: high-temporal-resolution observations for a volume scan of 30 s in a 60 km range, high-density observations below a 15 km altitude, improved rain rate estimation using dual-polarimetric observations, and hydrometer classification abilities. To achieve high-temporal-resolution observations, wide transmitted waves with a beam width up to 5 degrees are used for elevation angles. Although Fourier-domain digital beamforming can be used for receiving waves, the high sidelobe level of the antenna pattern using this type of beamforming results in substantial errors (e.g., overestimation of received power) because of ground clutter echoes. To solve this problem, an adaptive beamforming method based on the minimum mean square error (MMSE) was developed for a single-polarization phased-array radar. In this study, signal processing procedures, including the steering vector correction of real measurement data, are developed to apply the MMSE method to the DP-PAWR. The effect of phase errors and ground clutter on the dual-polarimetric parameters is evaluated via numerical simulations. Subsequently, the proposed method is applied to real measurement data of the DP-PAWR. Consequently, it is found that the clutter suppression achieved using the proposed method is clearly superior to that achieved using Fourier-domain digital beamforming.