The magnetization, M(H< or =30 T,0.7< or =T< or =300 K), of (C5H12N)2CuBr4 has been used to identify this system as an S = 1/2 Heisenberg two-leg ladder in the strong-coupling limit, J( perpendicular) = 13.3 K and J( parallel) = 3.8 K, with H(c1) = 6.6 T and H(c2) = 14.6 T. An inflection point in M(H,T = 0.7 K) at half saturation, M(s)/2, is described by an effective XXZ chain. The data exhibit universal scaling behavior in the vicinity of H(c1) and H(c2), indicating that the system is near a quantum critical point.