In this work we explore a new cosmological solution for an universe filled with one dissipative fluid, described by a barotropic equation of state (EoS) p=ωρ, in the framework of the full Israel-Stewart theory. The form of the bulk viscosity has been assumed of the form ξ=ξ0ρ1/2. The relaxation time is taken to be a function of the EoS, the bulk viscosity and the speed of bulk viscous perturbations, cb. The solution presents an initial singularity, where the curvature scalar diverges as the scale factor goes to zero. Depending on the values for ω, ξ0, cb accelerated and decelerated cosmic expansion can be obtained. In the case of accelerated expansion, the viscosity drives the effective EoS to be of quintessence type, for the single fluid with positive pressure. Nevertheless, we show that only the solution with decelerated expansion satisfies the thermodynamics conditions dS/dt>0 (growth of the entropy) and d2S/dt2<0 (convexity condition). We show that an exact stiff matter EoS is not allowed in the framework of the full causal thermodynamic approach; and in the case of a EoS very close to the stiff matter regime, we found that dissipative effects becomes negligible so the entropy remains constant. Finally, we show numerically that the solution is stable under small perturbations.