Recently, we have theoretically demonstrated that passive gravitational mass of a composite quantum body is not equivalent to its inertial mass due to some quantum effects. As an example, we have considered the simplest composite quantum body - a hydrogen atom. It is important that we have also suggested an idealized experiment in the Earth's orbit to observe the above mentioned phenomenon. Here, we discuss how to perform the corresponding real experiment. In particular, we consider He-4 atoms, solid He-4 under pressure, and crystalline dielectric solids. In the case where spacecraft moves them in the Earth's gravitational field with constant velocity, one can observe a breakdown of the Equivalent Principle by absorbing unusual radiation, emitted by one of the above mentioned macroscopic ensembles of the atoms. In the case of crystalline dielectric, one can observe a finite electric conductivity in the Earth's orbit. If one of such experiments is done, it will be not only the first observation of a breakdown of the Equivalence Principle, but also the first direct observation of quantum effects in General Relativity.