  Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$
all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this
paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for
$\frac{8}{9}n+112\le m\le \lceil\frac{3n}{2}\rceil+1$ and $n \geq 1000$. This
answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey
numbers, {\it Ars Combin.,} {\bf 31} (1991), 239--248) in a stronger form when
$m$ and $n$ are large. Building upon this exact result, we are able to
determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k \geq 3$.
Namely, we prove that for each $k \geq 3$, $r(B_n^{(k)}, C_n)= (k+1+o_k(1))n.$
This extends a result due to Rousseau and Sheehan (A class of Ramsey problems
involving trees, {\it J.~London Math.~Soc.,} {\bf 18} (1978), 392--396).
