A search for the rare decays Bs0→μ+μ- and B0→μ+μ- is performed at the LHCb experiment using data collected in p p collisions corresponding to a total integrated luminosity of 4.4 fb-1. An excess of Bs0→μ+μ- decays is observed with a significance of 7.8 standard deviations, representing the first observation of this decay in a single experiment. The branching fraction is measured to be B (Bs0→μ+μ- )=(3.0 ±0. 6-0.2+0.3) ×10-9 , where the first uncertainty is statistical and the second systematic. The first measurement of the Bs0→μ+μ- effective lifetime, τ (Bs0→μ+μ- )=2.04 ±0.44 ±0.05 ps , is reported. No significant excess of B0→μ+μ- decays is found, and a 95% confidence level upper limit, B (B0 →μ+μ-) <3.4 ×10-10 , is determined. All results are in agreement with the standard model expectations.