  We construct in $\mathbb{R}^{k+2}$ a $k$-dimensional simple normal current
whose support is purely $2$-unrectifiable. The result is sharp because the
support of a normal current cannot be purely $1$-unrectifiable and a
$(k+1)$-dimensional normal current can be represented as an integral of
$(k+1)$-rectifiable currents. This gives a negative answer to the (revised
version) of a question of Frank Morgan (1984).
