The magnetic exchange in hole-doped ferromagnetic cobaltates is investigated by studying the magnetic and electronic properties of La0.7Sr0.3CoO3 films as a function of epitaxial strain. We found a strong-coupling double exchange mechanism between Co3+ (4t(2g) 2e(g)) and Co4+ (3t(2g) 2e(g)) high-spin states mediated by t(2g) electrons--in contrast to the moderate coupling provided by the e(g) exchange in manganites. The strong sensitivity of the Curie temperature TC to the bulk compression can be explained by the small bandwidth of the t(2g)-derived states. A strain-induced Jahn-Teller effect is likewise observed. The experimental results clarify the magnetic exchange mechanism in the cobaltates.