Heat equation with heat source in form of delta function

Question

Consider the problem \begin{equation} \left\{\begin{array}{cc}u_t-u_{xx}=\delta_0,&0<x<1,\ t>0\\ u_x(0,t)=u_x(1,t)=0,&t>0,\\ u(x,0)=0,& 0\leq x\leq 1.\end{array}\right. \end{equation} Using Fourier transform one finds out that $$\hat u(\xi,t)=\frac{1}{\xi}\left(1-e^{-\xi t}\right),$$ but how can I relate this to the solution of the original problem?