The 1.28 GHz MeerKAT DEEP2 Image

We present the confusion-limited 1.28 GHz MeerKAT DEEP2 image covering one $\approx 68'$ FWHM primary beam area with $7.6''$ FWHM resolution and $0.55 \pm 0.01$ $\mu$Jy/beam rms noise. Its J2000 center position $\alpha=04^h 13^m 26.4^s$, $\delta=-80^\circ 00' 00''$ was selected to minimize artifacts caused by bright sources. We introduce the new 64-element MeerKAT array and describe commissioning observations to measure the primary beam attenuation pattern, estimate telescope pointing errors, and pinpoint $(u,v)$ coordinate errors caused by offsets in frequency or time. We constructed a 1.4 GHz differential source count by combining a power-law count fit to the DEEP2 confusion $P(D)$ distribution from $0.25$ to $10$ $\mu$Jy with counts of individual DEEP2 sources between $10$ $\mu$Jy and $2.5$ mJy. Most sources fainter than $S \sim 100$ $\mu$Jy are distant star-forming galaxies obeying the FIR/radio correlation, and sources stronger than $0.25$ $\mu$Jy account for $\sim93\%$ of the radio background produced by star-forming galaxies. For the first time, the DEEP2 source count has reached the depth needed to reveal the majority of the star formation history of the universe. A pure luminosity evolution of the 1.4 GHz local luminosity function consistent with the Madau & Dickinson (2014) model for the evolution of star-forming galaxies based on UV and infrared data underpredicts our 1.4 GHz source count in the range $-5 \lesssim \log[S(\mathrm{Jy})] \lesssim -4$.