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 thermometer, 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 compared to conventional four-wave mixing KIT devices. It is built in an artificial transmission line intrinsically matched to 50 Ω, whose dispersion allows for a controlled amplification bandwidth. We experimentally measure 17.6 +1.1 −1.4 dB of gain across a 2 GHz bandwidth, with an input 1 dB compression power of -63 dBm within that bandwidth, in qualitative agreement with theory. Using the KIT as the first amplifier in an amplification chain, we measure a system-added noise of 0.61±0.08 K between 3.5 and 5.5 GHz, about one eighth the noise obtained when using only a representative classical amplifier. The KIT contribution to this added noise is estimated to be 0.2 ± 0.1 K, consistent with the quantum limit on amplifier added noise. This device is therefore suitable to read large arrays of microwave kinetic inductance detectors or thousands of superconducting qubits.