  A $n\times n$ matrix $A$ has normal defect one if it is not normal, however
can be embedded as a north-western block into a normal matrix of size
$(n+1)\times (n+1)$. The latter is called a minimal normal completion of $A$. A
construction of all matrices with normal defect one is given. Also, a simple
procedure is presented which allows one to check whether a given matrix has
normal defect one, and if this is the case -- to construct all its minimal
normal completions. A characterization of the generic case for each $n$ under
the assumption $\rank(A^*A-AA^*)=2$ (which is necessary for $A$ to have normal
defect one) is obtained. Both the complex and the real cases are considered. It
is pointed out how these results can be used to solve the minimal commuting
completion problem in the classes of pairs of $n\times n$ Hermitian (resp.,
symmetric, or symmetric/antisymmetric) matrices when the completed matrices are
sought of size $(n+1)\times(n+1)$. An application to the $2\times n$
separability problem in quantum computing is described.
