************************************************************************* 63017 In a recent compilation of ROSAT detected radio-quiet quasars a small group of six nearby objects was found showing extremely low X-ray luminosities (by a factor of $\\sim$ 30) compared to the bulk of other quasars at similar optical luminosities. \\hfill \\break We propose to observe the most interesting of these objects, PG 0844+349, which showed, as the only object of this sample, in ROSAT observations time variability by a factor of six, for 60 ks with ASCA. Although the object has been studied extensively at longer wavelengths its X-ray properties are poorly known. The broad band ASCA spectrum and the temporal behavior of its X-ray emission promise to yield valuable information on the physical nature of this unusual, possibly rare class of objects. *************************************************************************