When convolution of $L^1$ function with $C_0$ function is $L^1$

For simplicity, let's only consider "signals", functions $f(t)$ on $\mathbb R$ that are identically zero on when $t<0$.

We know that it true that $L^1*C_0\subseteq C_0$ and $L^1*C_0\not\subseteq L^1$ where $C_0$ denotes continuous function with limit $0$.

Let $f\in L^1$ and $g\in C_0$. Now, I add another condition: $F(t):=\int_0^t f(s)ds$ is also in $L^1$. Then it seems to be true that $f*g(t)=\int_0^tf(t-s)g(s)ds$ is in $L^1$? I could not prove or find counter examples, but intuitively it seems right.