A search for charge-parity (
C
P
) violation in
D
0
→
K
−
K
+
and
D
0
→
π
−
π
+
decays is reported, using
p
p
collision data corresponding to an integrated luminosity of
5.9
 
 
fb
−
1
collected at a center-of-mass energy of 13 TeV with the LHCb detector. The flavor of the charm meson is inferred from the charge of the pion in
D
∗
(
2010
)
+
→
D
0
π
+
decays or from the charge of the muon in
¯
B
→
D
0
μ
−
¯
ν
μ
X
decays. The difference between the
C
P
asymmetries in
D
0
→
K
−
K
+
and
D
0
→
π
−
π
+
decays is measured to be
Δ
A
C
P
=
[
−
18.2
±
3.2
(
stat
)
±
0.9
(
syst
)
]
×
10
−
4
for
π
-tagged and
Δ
A
C
P
=
[
−
9
±
8
(
stat
)
±
5
(
syst
)
]
×
10
−
4
for
μ
-tagged
D
0
mesons. Combining these with previous LHCb results leads to
Δ
A
C
P
=
(
−
15.4
±
2.9
)
×
10
−
4
, where the uncertainty includes both statistical and systematic contributions. The measured value differs from zero by more than 5 standard deviations. This is the first observation of
C
P
violation in the decay of charm hadrons.