  Correlated electrons in transition metal oxides (TMOs) exhibit a variety of
emergent phases. When TMOs are confined to a single-atomic-layer thickness,
experiments so far have shown that they usually lose diverse properties and
become insulators. In an attempt to extend the range of electronic phases of
the single-atomic-layer oxide, we search for a metallic phase in a
monolayer-thick epitaxial SrRuO$_3$ film. Combining atomic-scale epitaxy and
angle-resolved photoemission measurements, we show that the monolayer SrRuO$_3$
is a strongly correlated metal. Systematic investigation reveals that the
interplay between dimensionality and electronic correlation makes the monolayer
SrRuO$_3$ an incoherent metal with orbital-selective correlation. Furthermore,
the unique electronic phase of the monolayer SrRuO$_3$ is found to be highly
tunable, as charge modulation demonstrates an incoherent-to-coherent crossover
of the two-dimensional metal. Our work emphasizes the potentially rich phases
of single-atomic-layer oxides and provides a guide to the manipulation of their
two-dimensional correlated electron systems.
