Similar to the optimal-doped, weak-ferromagnetic (WFM induced by canted antiferromagnetism, T(subscript Curie)=131K) and superconducting (T(subscript c)=56K) RuSr2GdCu2O8, underdoped RuSr2EuCu2O8 (T(subscript Curie)=133K, T(subscript c)=36K) also exhibited a spontaneous vortex state (SVS) between 16K and 36K. The low field (±20G) superconducting hysteresis loop indicates a weak and narrow Meissner state region of average lower critical field B(superscript ave subscript cl) (T)=B(superscript ave subscript cl) (0)[1-(T/T(subscript SVS))^2], with B(superscript ave subscript cl) (0)=7G and T(subscript SVS)=16K. The vortex melting transition (T(subscript melting)=21K) below T(subscript c), obtained from the broad resistivity drop and the onset of the diamagnetic signal, indicates a vortex liquid region due to the coexistence and interplay between superconductivity and the WFM order. No visible jump in specific heat was observed near T(subscript c) for Eu-and Gd-compounds. Finally, with the baseline from the nonmagnetic Eu-compound, the specific heat data analysis confirms the magnetic entropy associated with antiferromagnetic ordering of Gd^3+(J=S=7/2) at 2.5K to be close to N(subscript A)k ln 8, as expected.