  We present the first relativistic calculations of the final phase of inspiral
of a binary system consisting of two stars built predominantely of strange
quark matter (strange quark stars). We study the precoalescing stage within the
Isenberg-Wilson-Mathews approximation of general relativity using a multidomain
spectral method. A hydrodynamical treatment is performed under the assumption
that the flow is either rigidly rotating or irrotational, taking into account
the finite density at the stellar surface -- a distinctive feature with respect
to the neutron star case. The gravitational-radiation driven evolution of the
binary system is approximated by a sequence of quasi-equilibrium configurations
at fixed baryon number and decreasing separation. We find that the innermost
stable circular orbit (ISCO) is given by an orbital instability both for
synchronized and irrotational systems. This constrasts with neutron stars for
which the ISCO is given by the mass-shedding limit in the irrotational case.
The gravitational wave frequency at the ISCO, which marks the end of the
inspiral phase, is found to be 1400 Hz for two irrotational 1.35 Msol strange
stars and for the MIT bag model of strange matter with massless quarks and a
bag constant B=60 MeV/fm^3. Detailed comparisons with binary neutrons star
models, as well as with third order Post-Newtonian point-mass binaries are
given.
