We report on the discovery of seven low-metallicity stars selected from the Hamburg/ESO Survey, six of which are extremely metal-poor (EMP, [Fe/H]{<=}-3.0), with four having [Fe/H]{<=}-3.5. Chemical abundances or upper limits are derived for these stars based on high-resolution (R~35000) Magellan/MIKE spectroscopy, and are in general agreement with those of other very and extremely metal-poor stars reported in the literature. Accurate metallicities and abundance patterns for stars in this metallicity range are of particular importance for studies of the shape of the metallicity distribution function of the Milky Way's halo system, in particular for probing the nature of its low-metallicity tail. In addition, taking into account suggested evolutionary mixing effects, we find that six of the program stars (with [Fe/H]{<=}-3.35) possess atmospheres that were likely originally enriched in carbon, relative to iron, during their main-sequence phases. These stars do not exhibit overabundances of their s-process elements, and hence may be, within the error bars, additional examples of the so-called CEMP-no class of objects.