  We present a combined experimental and theoretical study of the spin-1/2
compound CuP$_2$O$_6$ 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 $J_{2D}\sim$ 40 K in the layers is an order of magnitude
larger than $J_{1D}\sim$ 4 K in the chains. Below $T_N\sim$ 8 K, CuP$_2$O$_6$
develops long-range order (LRO), 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 layers, which is evidenced by
the persistent paramagnetic ESR signal below $T_N$. Compared to typical
quasi-2D magnets, the ordering temperature of CuP$_2$O$_6$ $T_N/J_{2D}\sim$ 0.2
is unusually low, showing that weakly coupled spins sandwiched between 2D
magnetic units effectively decouple these units and impede the long-range
ordering.
