Integration of the product of a compact supported convolution

I know that in general case we have $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(s)g(t-s) ds dt = \left( \int_{-\infty}^{+\infty} f(t) dt \right) \left( \int_{-\infty}^{+\infty}g(t) dt \right) $$as the integration of a convolution.

But, is this result still true if the support is not the whole domain ? I mean,is the following equality true ?

$$\int_{0}^{1}\int_{0}^{1} f(s)g(t-s) ds dt = \left( \int_{0}^{1} f(t) dt \right) \left( \int_0^{1}g(t) dt \right) $$

If the result is wrong, do we have a simplified form for $\int_{0}^{1}\int_{0}^{1} f(s)g(t-s) ds dt$ ?