A search for the rare decays
B
0
s
→
μ
+
μ
−
and
B
0
→
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
B
0
s
→
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
(
B
0
s
→
μ
+
μ
−
)
=
(
3.0
±
0.
6
+
0.3
−
0.2
)
×
10
−
9
, where the first uncertainty is statistical and the second systematic. The first measurement of the
B
0
s
→
μ
+
μ
−
effective lifetime,
τ
(
B
0
s
→
μ
+
μ
−
)
=
2.04
±
0.44
±
0.05
 
 
ps
, is reported. No significant excess of
B
0
→
μ
+
μ
−
decays is found, and a 95% confidence level upper limit,
B
(
B
0
→
μ
+
μ
−
)
<
3.4
×
10
−
10
, is determined. All results are in agreement with the standard model expectations.