Single-molecule spintronics investigates electron transport through magnetic molecules that have an internal spin degree of freedom(1). To understand and control these individual molecules it is important to read their spin state(2). For unpaired spins, the Kondo effect has been observed(3-8) as a low-temperature anomaly at small voltages. Here, we show that a coupled spin pair in a single magnetic molecule can be detected(9,10) and that a bias voltage can be used to switch between two states of the molecule. In particular, we use the mechanically controlled break-junction technique(11) to measure electronic transport through a single-molecule junction containing two coupled spin centres that are confined on two Co2+ ions. Spin-orbit configuration interaction methods are used to calculate the combined spin system, where the ground state is found to be a pseudo-singlet and the first excitations behave as a pseudo-triplet. Experimentally, these states can be assigned to the absence and occurrence of a Kondo-like zero-bias anomaly in the low-temperature conductance data, respectively. By applying finite bias, we can repeatedly switch between the pseudo-singlet state and the pseudo-triplet state.