  The longitudinal double spin asymmetry A_1^rho for exclusive leptoproduction
of rho^0 mesons, mu + N -> mu + N + rho, is studied using the COMPASS 2002 and
2003 data. The measured reaction is incoherent exclusive rho^0 production on
polarised deuterons. The Q^2 and x dependence of A_1^rho is presented in a wide
kinematical range: 3x10^-3 < Q^2 < 7 (GeV/c)^2 and 5x10^-5 < x < 0.05. The
presented results are the first measurements of A_1^rho at small Q2 (Q2 < 0.1
(GeV/c)^2) and small x (x < 3x10^-3). The asymmetry is in general compatible
with zero in the whole kinematical range.
