  The Meissner effect and the 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 as well as anomalous proximity effects have been detected but a
Meissner effect has never been observed. Here we have probed the local
diamagnetic response in the normal state of an underdoped La1.94Sr0.06CuO4
layer (up to 46 nm thick, critical temperature Tc' < 5 K) which was brought
into close contact with two nearly optimally doped La1.84Sr0.16CuO4 layers (Tc
\approx 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 well above Tc' but below Tc. 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; however, this
induced order is very sensitive to thermal excitation.
