A cycle $\mathscr{C}$ in a graph G is called extendable if there is another cycle in G which contains all the vertices of $\mathscr{C}$ and exactly one other vertex. A graph G is cycle-extendable if all its non-Hamilton cycles are extendable. A balanced incomplete block design with parameters v, k, and λ , written BIBD(v,k,λ) , is a pair $\mathscr{D} = (V,\mathscr{B})$ of sets, where V is a set of v elements and $\mathscr{B}$ is a set of k -subsets of V –-called blocks–-such that each pair of elements of V appears in exactly λ blocks in $\mathscr{B}$. The 0-block-intersection graph of a design $\mathscr{D} = (V,\mathscr{B})$ is the graph $\overline{G}_\mathscr{D}$ whose vertex set is $\mathscr{B}$ and in which two blocks are adjacent if and only if they do not intersect. An $A_1'$ cyclic ordering of a BIBD(v,k,λ) is a listing of the elements of its block set such that consecutive blocks do not intersect and the last block does not intersect the first–-such an ordering corresponds to a Hamilton cycle in the 0-block-intersection graph of the design and to a 0-intersecting Gray code for the design, and vice versa. In this paper we demonstrate that if $\overline{G}_\mathscr{D}$ is the 0-block-intersection graph of $\mathscr{D}$ , a BIBD(v,k,λ) , then $\overline{G}_\mathscr{D}$ is Hamiltonian if λ = 1 and $v > \frac{1+\sqrt{5}}{2}k^2+k$, and cycle extendable if $v>2k^2+1$. We also present a polynomial-time algorithm for finding cycles of any length in the 0-block-intersection graph of a BIBD(v,k,λ) with $v > (2+\sqrt{3})k^2+1$ if λ = 1 or $v>5k^2+1$ λ ⩾ 2 .