High-damping materials are widely used in engineering fields. In order to increase the precision of vibration control to different levels, high-damping materials with high-rigidity are required. This study attempts to develop a new high-damping high-rigidity material using ductile ceramics based on the Al2TiO5-MgTi2O5 system, which has many continuous microcracks along the grain boundaries. Ductile ceramics have high internal friction (Q (-1) = 0.01-0.037), but very low rigidity (< 10 GPa). The rigidity of Al2TiO5-MgTi2O5 ceramics was improved by combining them with a polymer such as acrylic resin. The Young's modulus and internal friction of the composites of Al2TiO5-MgTi2O5 ceramics and acrylic resin are investigated. They show high-damping capacity (Q (-1) = 0.03-0.04) with high rigidity (E = 50-60 GPa), and their properties depend on those of the polymer. Thus, the composites fabricated using the above method can serve as high-damping high-rigidity materials.