The 18O/17O isotopic ratio of oxygen is a crucial measure of the secular enrichment of the interstellar medium by ejecta from high-mass versus intermediate-mass stars. So far, however, there is a lack of data, particularly from the Galactic center (GC) region. Therefore, we have mapped typical molecular clouds in this region in the J = 1–0 lines of C18O and C17O with the Delingha 13.7 m telescope (DLH). Complementary pointed observations toward selected positions throughout the GC region were obtained with the IRAM 30 m and Mopra 22 m telescopes. C18O/C17O abundance ratios reflecting the 18O/17O isotope ratios were obtained from integrated intensity ratios of C18O and C17O. For the first time, C18O/C17O abundance ratios are determined for Sgr C (V ∼ −58 km s−1), Sgr D (V ∼ 80 km s−1), and the 13 complex (V ∼ 80 km s−1). Through our mapping observations, abundance ratios are also obtained for Sgr A (∼0 and ∼50 km s−1 component) and Sgr B2 (∼60 km s−1), which are consistent with the results from previous single-point observations. Our frequency-corrected abundance ratios of the GC clouds range from 2.58 ± 0.07 (Sgr D, V ∼ 80 km s−1, DLH) to 3.54 ± 0.12 (Sgr A, ∼50 km s−1). In addition, strong narrow components (line width less than 5 km s−1) from the foreground clouds are detected toward Sgr D (−18 km s−1), the 13 complex (−18 km s−1), and M+5.3−0.3 (22 km s−1), with a larger abundance ratio around 4.0. Our results show a clear trend of lower C18O/C17O abundance ratios toward the GC region relative to molecular clouds in the Galactic disk. Furthermore, even inside the GC region, ratios appear not to be uniform. The low GC values are consistent with an inside-out formation scenario for our Galaxy.