In 1979, Berhart and Kainen conjectured that graphs of fixed genus g greater than or equal to 1 have unbounded pagenumber. This proves that genus g graphs can be embedded in 0(g) pages, thus disproving the conjecture. An Omega(square root of g) lower bound is also derived. The first algorithm in the literature for embedding an arbitrary graph in a book with a non-trivial upper bound on the number of pages is presented. First, the algorithm computes the genus g of a graph using the algorithm of Filotti, Miller, Reif (1979), which is polynomial-time for fixed genus. Second, it applies an optimal-time algorithm for obtaining an 0(g)-page book embedding. We give separate book embedding algorithms for the cases of graphs embedded in orientable and nonorientable surfaces. An important aspect of the construction is a new decomposition algorithm, of independent interest, for a graph embedded on a surface. Book embedding has application in several areas, two of which are directly related to the results obtained: fault-tolerant VLSI and complexity theory.