How localized electrons interact with delocalized electrons is a central question to many problems in sold-state physics(1-3). The simplest manifestation of this situation is the Kondo effect, which occurs when an impurity atom with an unpaired electron is placed in a metal(2), At low temperatures a spin singlet state is formed between the unpaired localized electron and delocalized electrons at the Fermi energy, Theories predict(4-7) that a Kondo singlet should form in a single-electron transistor (SET), which contains a confined ＇droplet＇ of electrons coupled by quantum-mechanical tunnelling to the delocalized electrons in the transistor＇s leads, If this is so, a SET could provide a means of investigating aspects of the Kondo effect under controlled circumstances that are not accessible in conventional systems : the number of electrons can be changed from odd to even, the difference in energy between the localized state and the Fermi level can be tuned, the coupling to the leads can be adjusted, voltage differences can be applied to reveal non-equilibrium Kondo phenomena : and a single localized state can be studied rather than a statistical distribution. But for SETs fabricated previously, the binding energy of the spin singlet has been too small to observe Kondo phenomena, Ralph and Buhrman(8) have observed the Kondo singlet at a single accidental impurity in a metal point contact, but with only two electrodes and without control over the structure they were not able to observe all of the features predicted, Here we report measurements on SETs smaller than those made previously, which exhibit all of the predicted aspects of the Kondo effect in such a system.