The exclusive process e^{+}e^{-}→ΛΛ[over ¯], with Λ→pπ^{-} and Λ[over ¯]→p[over ¯]π^{+}, has been studied at sqrt[s]=2.396 GeV for measurement of the timelike Λ electric and magnetic form factors, G_{E} and G_{M}. A data sample, corresponding to an integrated luminosity of 66.9 pb^{-1}, was collected with the BESIII detector for this purpose. A multidimensional analysis with a complete decomposition of the spin structure of the reaction enables a determination of the modulus of the ratio R=|G_{E}/G_{M}| and, for the first time for any baryon, the relative phase ΔΦ=Φ_{E}-Φ_{M}. The resulting values are R=0.96±0.14(stat)±0.02(syst) and ΔΦ=37°±12°(stat)±6°(syst), respectively. These are obtained using the recently established and most precise value of the asymmetry parameter α_{Λ}=0.750±0.010 measured by BESIII. In addition, the cross section is measured with unprecedented precision to be σ=118.7±5.3(stat)±5.1(syst) pb, which corresponds to an effective form factor of |G|=0.123±0.003(stat)±0.003(syst). The contribution from two-photon exchange is found to be negligible. Our result enables the first complete determination of baryon timelike electromagnetic form factors.