  A new N=4 superconformal algebra (SCA) is presented. Its internal affine Lie
algebra is based on the seven-dimensional Lie algebra su(2)\oplus g, where g
should be identified with a four-dimensional non-reductive Lie algebra. Thus,
it is the first known example of what we choose to call a non-reductive SCA. It
contains a total of 16 generators and is obtained by a non-trivial
In\"on\"u-Wigner contraction of the well-known large N=4 SCA. The recently
discovered asymmetric N=4 SCA is a subalgebra of this new SCA. Finally, the
possible affine extensions of the non-reductive Lie algebra g are classified.
The two-form governing the extension appearing in the SCA differs from the
ordinary Cartan-Killing form.
