We present a combined experimental and theoretical study of the spin- compound CuPO that features a network of two-dimensional (2D) antiferromagnetic (AFM) square planes, interconnected via one-dimensional (1D) AFM spin chains. Magnetic susceptibility, high-field magnetization, and electron spin resonance (ESR) data, as well as microscopic density-functional band-structure calculations and subsequent quantum Monte Carlo simulations, show that the coupling K in the layers is an order of magnitude larger than K in the chains. Below K, CuPO develops long-range order, as evidenced by a weak net moment on the 2D planes induced by anisotropic magnetic interactions of Dzyaloshinsky-Moriya type. A striking feature of this 3D ordering transition is that the 1D moments grow significantly slower than the ones on the 2D units, which is evidenced by the persistent paramagnetic ESR signal below . Compared to typical quasi-2D magnets, the ordering temperature of CuPO is unusually low, showing that weakly coupled spins sandwiched between 2D magnetic units effectively decouple these units and impede the long-range ordering.