We present detailed low-temperature transport measurements of a single quantum dot formed in an InGaAs/InP heterostructure with a strong tunnel coupling to the source and drain leads. The conventional spin-1/2 Kondo effect is observed for the quantum dot in the N=9 charge state. By changing the voltages applied to the quantum dot barrier gates, we find a zero-field splitting of the Kondo resonance and a zero-bias differential conductance, which shows a nonmonotonic in-plane magnetic field and temperature dependence. Using a two-site Hubbard model, we show that the main observed features can be explained in terms of Kondo correlation effects resulting from the exchange interaction between two localized spins.