The development of quantum computing architectures from early designs and current noisy devices to fully fledged quantum computers hinges on achieving fault tolerance using quantum error correction(1-4). However, these correction capabilities come with an overhead for performing the necessary fault-tolerant logical operations on logical qubits (qubits that are encoded in ensembles of physical qubits and protected by error-correction codes)(5-8). One of the most resource-efficient ways to implement logical operations is lattice surgery(9-11), where groups of physical qubits, arranged on lattices, can be merged and split to realize entangling gates and teleport logical information. Here we report the experimental realization of lattice surgery between two qubits protected via a topological error-correction code in a ten-qubit ion-trap quantum information processor. In this system, we can carry out the necessary quantum non-demolition measurements through a series of local and entangling gates, as well as measurements on auxiliary qubits. In particular, we demonstrate entanglement between two logical qubits and we implement logical state teleportation between them. The demonstration of these operations-fundamental building blocks for quantum computation-through lattice surgery represents a step towards the efficient realization of fault-tolerant quantum computation. Two logical qubits are encoded in ensembles of four physical qubits through the surface code, then entangled by lattice surgery, which is a protocol for carrying out fault-tolerant operations.