  Spontaneous vortex phase (SVP) is an exotic quantum matter in which quantized
superconducting vortices form in the absence of external magnetic field.
Although being predicted theoretically nearly 40 years ago, its rigorous
experimental verification still appears to be lacking. Here we present
low-field magnetic measurements on single crystals of the iron-based
ferromagnetic superconductor Eu(Fe$_{0.91}$Rh$_{0.09}$)$_{2}$As$_{2}$ which
undergoes a superconducting transition at $T_\mathrm{sc}$ = 19.6 K followed by
a magnetic transition at $T_\mathrm{m}$ = 16.8 K. We observe a characteristic
first-order transition from a Meissner state within
$T_\mathrm{m}<T<T_\mathrm{sc}$ to an SVP below $T_\mathrm{m}$, under a magnetic
field approaching zero. Additional isothermal magnetization and ac
magnetization measurements at $T\ll T_\mathrm{sc}$ confirm that the system is
intrinsically in a spontaneous-vortex ground state. The unambiguous
demonstration of SVP in the title material lays a solid foundation for future
imaging and spectroscopic studies on this intriguing quantum matter.
