  An antimagic labelling of a graph $G$ is a bijection
$f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$
distinguish all vertices. A well-known conjecture of Hartsfield and Ringel
(1994) is that every connected graph other than $K_2$ admits an antimagic
labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \&
Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \&
Lyngsie (2017)) independently introduced the weaker notion of a local antimagic
labelling, where only adjacent vertices must be distinguished. Both sets of
authors conjectured that any connected graph other than $K_2$ admits a local
antimagic labelling. We prove this latter conjecture using the probabilistic
method. Thus the parameter of local antimagic chromatic number, introduced by
Arumugam et al., is well-defined for every connected graph other than $K_2$ .
