  Azimuthal asymmetry of vector-meson production in single-transversely
polarized proton-proton collisions ($p\up p$) is calculated in a model based on
the string fragmentation. The string is spanned by a valence quark of the
projectile scattered on the target. The asymmetry is generated only during
fragmentation of the scattered quark into hadrons. The obtained asymmetry of
the $\rho$ mesons is opposite in sign to that of pions. On the other hand, if
the asymmetry were generated during the quark scattering then the asymmetries
of the vector and of the pseudoscalar mesons would be close to each other.
