We present Atacama Large Millimeter Array observations at an angular resolution of 0farcs1–0farcs2 of the disk surrounding the young Herbig Ae star MWC 758. The data consist of images of the dust continuum emission recorded at 0.88 millimeter, as well as images of the 13CO and C18O J = 3–2 emission lines. The dust continuum emission is characterized by a large cavity of roughly 40 au in radius which might contain a mildly inner warped disk. The outer disk features two bright emission clumps at radii of ~47 and 82 au that present azimuthal extensions and form a double-ring structure. The comparison with radiative transfer models indicates that these two maxima of emission correspond to local increases in the dust surface density of about a factor 2.5 and 6.5 for the south and north clumps, respectively. The optically thick 13CO peak emission, which traces the temperature, and the dust continuum emission, which probes the disk midplane, additionally reveal two spirals previously detected in near-IR at the disk surface. The spirals seen in the dust continuum emission present, however, a slight shift of a few au toward larger radii and one of the spirals crosses the south dust clump. Finally, we present different scenarios to explain the complex structure of the disk.