The fractional differential equation \[ u_{tt}(t,x)=\int^t_0 k(t-s)u_{sxx}(s,x)\,ds+u_{xx}(t,x),\quad t>0,x\in(0,1)\tag{1} \] with boundary conditions (2) \(u(t,0)=u(t,1)=0\), and initial conditions (3) \(u(0,x)=u_0(x)\) and \(u_t(0,x)=u_1(x)\), \(x\in(0,1)\) is considered here. The kernel \(k(t)\) is taken in the form \(t^{-\alpha}e^{-\beta t}\), \(00\) and is called a weakly singular kernel. It is shown that the solution of the problem with a weakly singular kernel decays exponentially to zero.