  We construct a new version of the higher covariant derivative regularization
for a general ${\cal N}=2$ supersymmetric gauge theory formulated in terms of
${\cal N}=1$ superfields. This regularization preserves both supersymmetries of
the classical action, namely, the invariance under the manifest ${\cal N}=1$
supersymmetry and under the second hidden on-shell supersymmetry. The
regularizing ${\cal N}=2$ supersymmetric higher derivative term is found in the
explicit form in terms of ${\cal N}=1$ superfields. Thus, ${\cal N}=2$
supersymmetry is broken only by the gauge fixing procedure. Then we analyze the
exact NSVZ $\beta$-function and prove that in the considered model its higher
loop structure is determined by the anomalous dimension of the chiral
superfield $\Phi$ in the adjoint representation which is the ${\cal N}=2$
superpartner of the gauge superfield $V$. Using the background field method we
find that this anomalous dimension is related with the anomalous dimension of
the hypermultiplet and vanishes if the effective action is invariant under
${\cal N}=2$ background supersymmetry. As a consequence, in this case the
higher loop contributions to $\beta$-function also vanish. The one-loop
renormalization structure in the considered regularization is also studied by
the explicit calculations of the one-loop renormalization constants.
