Cement mortar mixes were prepared to test the suitability of fine reclaimed asphalt pavement (FRAP) aggregates as a replacement for natural fine aggregates (NFA). With different amounts of NFA (25%, 50%, 75% and 100% by weight) replaced with FRAP aggregates, the mechanical qualities of the mortar mix deteriorated. To counteract this, jarosite – a waste from the zinc industry – was used as a partial replacement for cement (replacements of 5%, 10% and 15%). The jarosite improved the microstructural, mechanical and shrinkage properties. For example, compared with the control mix (100% NFA), the mix containing 25% FRAP aggregates showed a 5.5% reduction in hardened density and a 14% reduction in compressive strength. With the addition of 10% jarosite, there was an increase of 4.85% in hardened density and 7% in compressive strength with respect to the mix containing 25% FRAP aggregates. It is thus suggested that 10% jarosite is used when FRAP aggregates are used to make cement mortar mixes. A cost analysis was also performed. The findings of this work are expected to inspire rational mix design recommendations for mortar mixes including FRAP aggregates, as well as bringing environmental and economic benefits by reducing carbon dioxide footprints.