We measure the branching ratio of the purely leptonic decay of the D+ meson with unprecedented precision as B(D+ -> mu(+nu)) = (3.82 +/- 0.32 +/- 0.09) x 10(-4), using 818 pb(-1) of data taken on the psi(3770) resonance with the CLEO-c detector at the CESR collider. We use this determination to derive a value for the pseudoscalar decay constant f(D+), combining with measurements of the D+ lifetime and assuming vertical bar V-cd vertical bar = vertical bar V-us vertical bar. We find f(D+) = (205.8 +/- 8.5 +/- 2.5) MeV. The decay rate asymmetry Gamma(D+ -> mu(+)nu) - Gamma(D- -> mu(-) (nu) over bar) / Gamma(D+ -> mu(+)nu) + Gamma(D- -> mu(-) (nu) over bar) = 0.08 +/- 0.08, consistent with no CP violation. We also set 90% confidence level upper limits on B(D+ -> tau(+) v) < 1.2 x 10(-3) and B(D+ -> e(+) v) < 8.8 x 10(-6).