A new Cu-related photoluminescence band in InP is reported. It consists of a sharp zero-phonon line at 1.2889 eV with sharp LO and 'gap mode' local phonon replicas to lower energy superimposed on a broad vibronic background. Zeeman studies show that the recombination process arises from an exciton bound to a neutral isoelectronic centre. Electron and hole g-values of 1.27 and 2.12 respectively are deduced, with both electron and hole having spin ¹/₂. The very strong diamagnetic shift of the spectrum is found to be equal to that of a shallow donor electron in InP. This demonstrates clearly that the hole is tightly bound in the short-range potential of the centre with binding energy E_B^h=127.5 meV and that the electron is then weakly bound in the Coulomb field of the hole (E_B^e=7.3 meV), so that the overall behaviour is of an isoelectronic donor. The thermal activation energy of the vibronic sideband of the luminescence is found to be 135+or-10 meV, in good agreement with the spectroscopic value of the hole binding energy.