  We analyze in detail the relation between an exactly marginal deformation of
N=4 SYM - the Leigh-Strassler or ``beta-deformation'' - and its string theory
dual (recently constructed in hep-th/0502086) by comparing energies of
semiclassical strings to anomalous dimensions of gauge-theory operators in the
two-scalar sector. We stress the existence of integrable structures on the two
sides of the duality. In particular, we argue that the integrability of strings
in AdS_5 x S^5 implies the integrability of the deformed world sheet theory
with real deformation parameter. We compare the fast string limit of the
worldsheet action in the sector with two angular momenta with the continuum
limit of the coherent state action of an anisotropic XXZ spin chain describing
the one-loop anomalous dimensions of the corresponding operators and find a
remarkable agreement for all values of the deformation parameter. We discuss
some of the properties of the Bethe Ansatz for this spin chain, solve the Bethe
equations for small number of excitations and comment on higher loop properties
of the dilatation operator. With the goal of going beyond the leading order in
the 't Hooft expansion we derive the analog of the Bethe equations on the
string-theory side, and show that they coincide with the thermodynamic limit of
the Bethe equations for the spin chain. We also compute the 1/J corrections to
the anomalous dimensions of operators with large R-charge (corresponding to
strings with angular momentum J) and match them to the 1-loop corrections to
the fast string energies. Our results suggest that the impressive agreement
between the gauge theory and semiclassical strings in AdS_5 x S^5 is part of a
larger picture underlying the gauge/gravity duality.
