We present a theoretical model and experimental characterization of a microwave kinetic inductance traveling-wave amplifier (KIT), whose noise performance, measured by a shot-noise tunnel junction (SNTJ), approaches the quantum limit. Biased with a dc current, the KIT operates in a three-wave mixing fashion, thereby reducing by several orders of magnitude the power of the microwave pump tone and associated parasitic heating compared to conventional four-wave mixing KIT devices. It consists of a 50 Ohms artificial transmission line whose dispersion allows for a controlled amplification bandwidth. We measure $16.5^{+1}_{-1.3}$ dB of gain across a 2 GHz bandwidth with an input 1 dB compression power of -63 dBm, in qualitative agreement with theory. Using a theoretical framework that accounts for the SNTJ-generated noise entering both the signal and idler ports of the KIT, we measure the system-added noise of an amplification chain that integrates the KIT as the first amplifier. This system-added noise, $3.1pm0.6$ quanta (equivalent to $0.66pm0.15$ K) between 3.5 and 5.5 GHz, is the one that a device replacing the SNTJ in that chain would see. This KIT is therefore suitable to read large arrays of microwave kinetic inductance detectors and promising for multiplexed superconducting qubit readout.
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