Let $\F_q^n$ be a vector space of dimension $n$ over the finite field $\F_q$.
A $q$-analog of a Steiner system (briefly, a $q$-Steiner system), denoted
$S_q[t,k,n]$, is a set $S$ of $k$-dimensional subspaces of $\F_q^n$ such that
each $t$-dimensional subspace of $\F_q^n$ is contained in exactly one element
of $S$. Presently, $q$-Steiner systems are known only for $t=1$, and in the
trivial cases $t = k$ and $k = n$. Invthis paper, the first nontrivial
$q$-Steiner systems with $t >= 2$ are constructed. Specifically, several
nonisomorphic $q$-Steiner systems $S_2[2,3,13]$ are found by requiring that
their automorphism groups contain the normalizer of a Singer subgroup of
$\GL(13,2)$. This approach leads to an instance of the exact cover problem,
which turns out to have many solutions.