  We report the measurements of resistivity and magnetization under magnetic
fields parallel and perpendicular to the basal plane, respectively, on a
cobalt-doped Eu(Fe$_{0.89}$Co$_{0.11}$)$_{2}$As$_{2}$ single crystal. We
observed a resistivity drop at $T_c\sim$ 21 K, which shifts toward lower
temperatures under external fields, suggesting a superconducting transition.
The upper critical fields near $T_c$ show large anisotropy, in contrast with
those of other '122' FeAs-based superconductors. Low-field magnetic
susceptibility data also show evidence of superconductivity below 21 K. Instead
of expected zero-resistance below $T_c$, however, a resistivity reentrance
appears at 17 K under zero field, coincident with the magnetic ordering of
Eu$^{2+}$ moments. Based on the temperature and field dependences of
anisotropic magnetization, a helical magnetic structure for the Eu$^{2+}$ spins
is proposed. External magnetic fields easily changes the helimagnetism into a
ferromagnetism with fully polarized Eu$^{2+}$ spins, accompanying by
disappearance of the resistivity reentrance. Therefore, superconductivity
coexists with ferromagnetic state of Eu$^{2+}$ spins under relatively low
magnetic field. The magnetic and superconducting phase diagrams are finally
summarized for both $H\parallel ab$ and $H\parallel c$.
