  Let $V$ be a vector space over the finite field ${\mathbb F}_q$. A
$q$-Steiner system, or an $S(t,k,V)_q$, is a collection ${\mathcal B}$ of
$k$-dimensional subspaces of $V$ such that every $t$-dimensional subspace of
$V$ is contained in a unique element of ${\mathcal B}$. A large set of
$q$-Steiner systems, or an $LS(t,k,V)_q$, is a partition of the $k$-dimensional
subspaces of $V$ into $S(t,k,V)_q$ systems. In the case that $V$ has infinite
dimension, the existence of an $LS(t,k,V)_q$ for all finite $t,k$ with $1<t<k$
was shown by Cameron in 1995. This paper provides an explicit construction of
an $LS(t,t+1,V)_q$ for all prime powers $q$, all positive integers $t$, and
where $V$ has countably infinite dimension.
