[en] DyTe_3 is a quasi-two-dimensional system in which two successive incommensurate charge-density-wave (CDW) states appear upon cooling at ambient pressure (T_C_D_W_,_1 = 306 K, T_C_D_W_,_2 = 49 K). The suppression with pressure of the CDW order is followed by the emergence of superconductivity above 1 GPa and below 1.5 K, as shown by our measurements of electrical resistivity and ac-susceptibility. X-ray diffraction (XRD) experiments under pressure indicate that the lower CDW state merges with the upper one at an intermediate pressure, suggesting that the double-CDW state could be accessed directly below a single T_C_D_W(P) line. The phase diagram obtained from XRD is compared with the results of our recent electrical resistivity experiments.