We report a novel aspect of the competition and coexistence between magnetism and superconductivity in the high-T(c) cuprate La(2-x)Sr(x)CuO4 (La214). With a modest magnetic field applied H parallel c axis, we monitored the infrared signature of pair tunneling between the CuO2 planes and discovered the complete suppression of interlayer coupling in a series of underdoped La214 single crystals. We find that the in-plane superconducting properties remain intact, in spite of enhanced magnetism in the planes.