  Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector
subspaces of $\mathbb R^n$. Using the Hodgkin spectral sequence, we compute the
complex $K$-ring of $G_{n,k}$, up to a small indeterminacy, for all values of
$n,k$ where $2\le k\le n-2$. When $n\equiv 0\!\!\mod 4, k\equiv 1\!\!\mod 2$
our result is complete.
