The paper agreed to training in reporting suicide sensitively, to be conducted by SamaritansCredit: Lewis Stickley/PA
The Press Complaints Commission negotiated the deal between the paper and the charities, who had complained to the PCC about the "flippant" tone and graphic imagery used in the Clarkson article.
The complaint was made under clause five of the editors' code, which relates to intrusion into grief or shock. The Sun accepted that parts of the column had overstepped the mark and apologised for the offence caused.
The matter was resolved after the paper agreed to training in reporting suicide sensitively, to be conducted by Samaritans.
It has also agreed to support the Time to Change campaign against mental health discrimination led by the charities Mind and Rethink Mental Illness and to offer training to its staff on mental health reporting.
PCC head of complaints Charlotte Dewar said in a release: "This is a good example of how a complaint to the PCC can lead to a positive outcome for both parties.
"We are pleased to have assisted the charities in resolving this issue, and we hope this agreement will be the basis for constructive engagement between the Sun and all of the charities involved."