The author studies the symmetric random block matrices \(\Xi_{pq\times pq}=\{\delta_{ij}A_{q\times q}+ {1\over\sqrt{p}} \Xi_{q\times q}^{(ij)}\}_{i,j=1}^{p}\), where \(\delta_{ij}\) is a Kronecker symbol, \(A_{q\times q}\) are nonrandom matrices, \(\|A_{q\times q}\|\leq c<\infty\), \(\Xi_{q\times q}^{(ij)}\), \(i\geq j\), \(i,j=1,\ldots, p\), are independent random matrices with distribution \(P\{\Xi_{q\times q}^{(ij)}= B_{q\times q}\}= P\{\Xi_{q\times q}^{(ij)}=- B_{q\times q}\}=1/2\), \(i,j=1,\ldots,p\), \(p=1,2,\ldots\), and \(B_{q\times q}\) is a real symmetric positive defined matrix, \(\|B_{q\times q}\|\leq c<\infty\). The author proves that for almost all \(x\) with probability one \(\lim_{p,q\to\infty}|\mu_{pq}(x,\Xi_{pq\times pq}) -F_{q}(x)|=0\), where \(\mu_{pq}(x,\Xi_{pq\times pq})\) is a normalized spectral function and \(F_{q}(x)\) is the distribution function whose block matrix density is presented in explicit form. For some matrices \(A_{q\times q}\) and \(B_{q\times q}\) the block matrix density of \(F_{q}(x)\) is equal to the sum of the semicircle laws.