The axion emerges in extensions of the Standard Model that explain the absence of CP violation in the strong interactions. Simultaneously, it can provide naturally the cold dark matter in our universe. Several searches for axions and axion-like particles (ALPs) have constrained the corresponding parameter space over the last decades but no unambiguous hints of their existence have been found. The axion mass range below 1 meV remains highly attractive and a well motivated region for dark matter axions. In this White Paper we present a description of a new experiment based on the concept of a dielectric haloscope for the direct search of dark matter axions in the mass range of 40 to 400 Formula omitted. This MAgnetized Disk and Mirror Axion eXperiment (MADMAX) will consist of several parallel dielectric disks, which are placed in a strong magnetic field and with adjustable separations. This setting is expected to allow for an observable emission of axion induced electromagnetic waves at a frequency between 10 to 100 GHz corresponding to the axion mass.