The judge at the five-day trial at Leeds Crown Court said Mr King displayed a "gross breach of trust".
BBC: Fatal helicopter flight instructor Ian King found guilty
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"Both these offences constitute a grossbreach of trust, " said the judge.
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But prosecutor Mark Bryant-Heron said Casburn was guilty of a "gross breach" of public trust and had "sought to undermine a highly sensitive and high-profile investigation".
BBC: April Casburn guilty of leak to News of the World offer