Let ∧ = {λ1,...,λnk} be amultisetofelements ofanintegral domain R.Let P be a partially prescribed n X n block matrix such that each prescribed entry is a k—block (a k X k matrix over R). If P has at most 2n — 3 prescribed entries then the unprescribed entries of P can be filled with k—blocks to obtain a matrix over R with spectrum ∧ (some natural conditions on the prescribed entries are required). We describe an algorithm to construct such completion.