The design, R&D, and testing of a new plastic-scintillator detector for Time-of-Flight measurements with relativistic heavy-ion beams are presented. A design approach using 32 independent precise timing measurements of the same physical event is followed. This is different from the conventional scheme, which aims at two or four high-precision measurements. A circular, 27 cm in diameter, BC-420 plastic-scintillator sheet is read-out by 32 photomultiplier tubes in order to achieve an intrinsic detector resolution on the order of 10 ps root mean square.