Magnetization and specific heat of an amorphous Dy-Cu alloy containing 41-at.% dysprosium establish that random single-ion anisotropy dominates exchange in this material, which can be approximately considered as a random Ising magnet. Exchange coupling is predominantly ferromagnetic with some antiferromagnetic interactions. Square hysteresis loops at 100 mK which show large values of remanence and giant Barkhausen jumps indicate that the magnetic structure in the magnetized state is asperomagnetic (random ferromagnetic). Remanence and coercivity fall off exponentially with increasing temperature from limiting values of 2.7B/Dy and 8000 Oe. Time dependences were studied both of the remanence, which decays with the logarithm of time, and the magnetization in low fields which increases exponentially with time below 18 K when the temperature is increased with a time constant practically independent of temperature. The spin-freezing transition at Tf=18 K is marked by a peak in the irreversible low-field magnetization measured in a SQUID magnetometer but there is no marked anomaly in the specific heat. An attempt to analyze the data near Tf in terms of critical exponents, justified by the large spontaneous magnetization, leads to values which differ from those of a normal ferromagnet, yet do not contradict the scaling-law equalities. © 1981 The American Physical Society.