When the ionosphere is illuminated with two strong HF signals, nonlinear mixing processes can take place near the region of reflection. The first direct experimental evidence of such processes occurring in ionospheric plasma are presented in this paper. The experiments were performed at Arecibo, where the HF ionospheric modifier was split to operate as two pure isolated and independent pumps over the frequency range of 3-8 MHz and the frequency difference between the two pumps was varied between a fraction of a Hz and several kHz. The purity of the transmitted signals was always ensured. Strong nonlinear effects were observed when the frequency difference was several Hz. Specifically, the HF signal from the ionosphere contained sidebands which were separated by multiples of the difference (Δf) between the two pumping frequencies. The first sidebands were typically 20--40 db below the pump signal. The strengths of the first sidebands increased, when the difference (Δf) between the two pumping frequencies decreased. Below about 5-10 Hz, the power in the first sidebands varied almost in Δf-n fashion, with n between 1 and 2. The power in the sidebands was also a strong nonlinear function of the incident pump field strength, showing the presence of a threshold. It is suggested that studies of such nonlinear mixing processes would be valuable both for diagnostic and practical usage of ionospheric plasma.