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Holography has taught us that spacetime is emergent and its properties depend on the entanglement structure of the dual theory. In this paper, we describe how changes in the entanglement due to a local projective measurement (LPM) on a subregion $A$ of the boundary theory modify the bulk dual spacetime. We find that LPMs destroy portions of the bulk geometry, yielding post-measurement bulk spacetimes dual to the complementary unmeasured region $A^c$ that are cut off by end-of-the-world branes. Using a bulk calculation in $AdS_3$ and tensor network models of holography, we show that the portions of the bulk geometry that are preserved after the measurement depend on the size of $A$ and the state we project onto. The post-measurement bulk dual to $A^c$ includes regions that were originally part of the entanglement wedge of $A$ prior to measurement. This suggests that LPMs performed on a boundary subregion $A$ teleport part of the bulk information originally encoded in $A$ into the complementary region $A^c$. In semiclassical holography an arbitrary amount of bulk information can be teleported in this way, while in tensor network models the teleported information is upper-bounded by the amount of entanglement shared between $A$ and $A^c$ due to finite-$N$ effects. When $A$ is the union of two disjoint subregions, the measurement triggers an entangled/disentangled phase transition between the remaining two unmeasured subregions, corresponding to a connected/disconnected phase transition in the bulk description. Our results shed new light on the effects of measurement on the entanglement structure of holographic theories and give insight on how bulk information can be manipulated from the boundary theory. They could also be extended to more general quantum systems and tested experimentally, and represent a first step towards a holographic description of measurement-induced phase transitions.