Since the formal errors of many geodetic time series are also available, this paper proposes a new singular spectrum analysis (SSA) approach with stabilizing weights (SSAW) by taking the formal errors into account, where the weight of time series data is constructed based on the ratio of formal error to signal power spectrum. The formulae of the proposed SSAW are derived in detail and then used to process the real Global Mean Sea Level (GMSL) time series compared to the traditional SSA. When the first 10 principal components are used to fit the GMSL time series, the fitting errors of the SSAW and traditional SSA are 4.80 mm and 5.14 mm, with the reduction of 6.61 %. According to the 500 simulations based on the reconstructed signals and formal errors of GMSL time series, the mean root mean squared errors and mean absolute errors of reconstructed signals using the SSAW relative to traditional SSA are reduced from 2.18 mm to 1.66 mm and 1.67 mm to 1.34 mm, respectively. Therefore, if the formal errors of a noisy time series given, the proposed SSAW approach is suggested to analyze this time series.