  Ordered electronic states have been extensively explored in cuprates and
iron-based unconventional superconductors, but seldom observed in the epitaxial
FeSe/SrTiO3(001) monolayer (FeSe/STO) with an enhanced superconducting
transition temperature (Tc). Here, by using scanning tunneling microscopy/
spectroscopy (STM/STS), we reveal a checkerboard charge order in the epitaxial
FeSe/STO monolayer, with a period of four times the inter-Fe-atom distance
along two perpendicular directions of the Fe lattice. This ordered state is
uniquely present in the superconducting FeSe/STO monolayer, even at liquid
nitrogen temperature, but absent in the non-superconducting FeSe monolayer or
bilayer. Quasiparticle interference (QPI) measurements further confirm it as a
static order without an energy-dependent dispersion and gapped out within the
superconductivity gap. The intensity of the charge order shows an enhancement
near the superconducting transition temperature, thus implying a correlation
with the high-Tc superconductivity in the FeSe/STO monolayer. This study
provides a new basis for exploring the ordered electronic states and their
interplay with high-Tc superconductivity in the FeSe monolayer.
