The Meissner effect and associated perfect 'bulk' diamagnetism together with zero resistance and gap opening are characteristic features of the superconducting state. In the pseudogap state of cuprates, unusual diamagnetic signals and anomalous proximity effects have been detected, but a Meissner effect has never been observed. Here we probe the local diamagnetic response in the normal state of an underdoped La(1.94)Sr(0.06)CuO(4) layer (T(c)'≤5 K), which is brought into close contact with two nearly optimally doped La(1.84)Sr(0.16)CuO(4) layers (T(c)≈32 K). We show that the entire 'barrier' layer of thickness, much larger than the typical c axis coherence lengths of cuprates, exhibits a Meissner effect at temperatures above T(c)' but below T(c). The temperature dependence of the effective penetration depth and superfluid density in different layers indicates that superfluidity with long-range phase coherence is induced in the underdoped layer by the proximity to optimally doped layers, but this induced order is sensitive to thermal excitation.