A new class of magnetic behavior, random quantum spin chain paramagnetism, has been observed in the one-dimensional compound Sr3CuPt1-xIrxO6. A random quantum spin chain system has S = 1/2 spins coupled by Heisenberg exchange interactions that are randomly ferromagnetic or antiferromagnetic between neighbors along the chain. This condition was fulfilled by members of the solid solution Sr3CuPt1-xIrxO6 (x = 0, 0.25, 0.50, 0.75, and 1), whose end-members, Sr3CuPtO6 and Sr3CuIrO6, are antiferromagnetic and ferromagnetic, respectively. Magnetic susceptibility data for the solid solution Sr3CuPt1-xIrxO6 (x = 0, 0.25, 0.50, 0.75, and 1) were collected and were found to be in excellent agreement with a theoretical model.