We present Atacama Large Millimeter and Submillimeter Array observations of the protoplanetary disk around the Herbig Ae star HD 163296 that trace the spatial distribution of millimeter-sized particles and cold molecular gas on spatial scales as small as 25 astronomical units (A.U.). The image of the disk recorded in the 1.3 mm continuum emission reveals three dark concentric rings that indicate the presence of dust depleted gaps at about 60, 100, and 160 A.U. from the central star. The maps of the 12CO, 13CO, and C 18O J =2 -1 emission do not show such structures but reveal a change in the slope of the radial intensity profile across the positions of the dark rings in the continuum image. By comparing the observations with theoretical models for the disk emission, we find that the density of CO molecules is reduced inside the middle and outer dust gaps. However, in the inner ring there is no evidence of CO depletion. From the measurements of the dust and gas densities, we deduce that the gas-to-dust ratio varies across the disk and, in particular, it increases by at least a factor 5 within the inner dust gap compared to adjacent regions of the disk. The depletion of both dust and gas suggests that the middle and outer rings could be due to the gravitational torque exerted by two Saturn-mass planets orbiting at 100 and 160 A.U. from the star. On the other hand, the inner dust gap could result from dust accumulation at the edge of a magnetorotational instability dead zone, or from dust opacity variations at the edge of the CO frost line. Observations of the dust emission at higher angular resolution and of molecules that probe dense gas are required to establish more precisely the origins of the dark rings observed in the HD 163296 disk.