Many solar prominences end their lives in eruptions or abrupt disappearances that are associated with dynamical or thermal instabilities. Such instabilities are important because they may be responsible for energy transport and conversion. We present a clear observation of a streaming kink-mode Kelvin-Helmholtz Instability (KHI) taking place in a solar prominence using the H alpha Lyot filter installed at the New Vacuum Solar Telescope, Fuxian-lake Solar Observatory in Yunnan, China. On one side of the prominence, a series of plasma blobs floated up from the chromosphere and streamed parallel to the limb. The plasma stream was accelerated to about 20-60 km s(-1) and then undulated. We found that 2 ''- and 5 ''-size vortices formed, floated along the stream, and then broke up. After the 5 ''-size vortex, a plasma ejection out of the stream was detected in the Solar Dynamics Observatory/Atmospheric Imaging Assembly images. Just before the formation of the 5 ''-size vortex, the stream displayed an oscillatory transverse motion with a period of 255 s with the amplitude growing at the rate of 0.001 s(-1). We attribute this oscillation of the stream and the subsequent formation of the vortex to the KHI triggered by velocity shear between the stream, guided by the magnetic field and the surrounding media. The plasma ejection suggests the transport of prominence material into the upper layer by the KHI in its nonlinear stage.