Facing a roomful of reporters and cameras, a visibly tired and grave Mr Palmer, accompanied by a solemn DPM Teo, announced his resignation.
Reading from a written statement, he told the press he had resigned "to avoid further embarrassment to the PAP and to Parliament".
He declined to take personal questions on his affair and how it had affected his family.
In his letter to the PM, also dated on Wednesday, he tendered his resignation as MP and as a PAP member with immediate effect.
He said he took full responsibility for his "grave mistake".
"My conduct was improper and it was a serious error of judgment," he wrote.
PM Lee accepted the resignation "with great sadness" yesterday.
He said: "It is necessary that all PAP MPs and advisers to grassroots organisations uphold the highest standards of personal conduct... and that the party be seen to hold its MPs and advisers to these standards."
Mr Palmer also asked for his family to be spared the publicity.
"I humbly request that you be kind enough to respect my family's privacy during this difficult period," he said.
This appeal was repeated by both DPM Teo and PM Lee. DPM Teo told reporters that those involved "are suffering much hurt from this episode".
"They need time and space for the healing to take place. I hope Singaporeans give them this time and space to do so, and that in time, they will rebuild their lives," he said.
The PA identified the woman as Madam Laura Ong, constituency director of Pasir Ris West Constituency Office, in a press statement on Wednesday evening.
It said she had tendered her resignation on Monday, citing family commitments.
On Wednesday night, Mr Palmer and his wife met grassroots leaders and volunteers of the Punggol East ward.
In a Facebook post, DPM Teo, who was also there with Mr Teo Ser Luck, said Mr Palmer had apologised to the grassroots members.
"There were tears in many eyes as they wished Michael and his family well," he wrote.
Mr Palmer did not respond to TNP's requests on Wednesay to comment on the SMS and e-mail exchanges.
Coincidentally, an American novelist named Michael Palmer released his new book this week. Its title - Political Suicide.
Get The New Paper for more stories.