The double spin asymmetry in the (-->)e(-->)p --> e(prime)pi(+)n reaction has been measured for the first time in the resonance region for four-momentum transfer Q2 = 0.35-1.5 GeV(2). Data were taken at Jefferson Lab with the CLAS detector using a 2.6 GeV polarized electron beam incident on a polarized solid NH3 target. Comparison with predictions of phenomenological models shows strong sensitivity to resonance contributions. Helicity-1/2 transitions are found to be dominant in the second and third resonance regions. The measured asymmetry is consistent with a faster rise with Q(2) of the helicity asymmetry A1 for the F(15)(1680) resonance than expected from the analysis of the unpolarized data.