On a new identity for double sum related to Bernoulli numbers

Let $m$, $n$ and $l$ be integers with $0\leqslant l\leqslant m+n$. It is the main purpose of this paper to give an identity for the sum:
$$\mathop{\sum_{a=0}^{m} \sum_{b=0}^{n}}_{a+b\geqslant m+n-l}B_{m-a}B_{n-b}\frac{\binom{m}{a}\binom{n}{b}}{a+b+1}\binom{a+b+1}{m+n-l},$$
where $B_m$ $(m=0,1,2,\dots)$ is the Bernoulli number. As corollary we prove that the above sum equal to $\dfrac{1}{2}$ when $l=0$.