The ex-No 10 spin doctor said he "failed to address" his own suspicions about the reporter, who was asking for the cash to pay the source.
Mr Coulson also repeatedly denied allegations he was aware of phone-hacking techniques being used by some News of the World (NotW) employees, and rejected a claim he had bullied co-defendant and ex royal editor Clive Goodman.
The payment, disclosed in an email exchange between Goodman and Mr Coulson in 2003, relates to a "green book" of contacts for royal staff.
In the email, Mr Goodman said he wanted to obtain the book for the "standard fee" of £1,000 from a royal policeman but the money should be paid in cash - something he needed Mr Coulson to sign for. The court heard Mr Goodman wrote that getting caught making the payment meant the police officer "could end up on criminal charges, as could we".
Mr Coulson, who said earlier that Goodman was prone to exaggeration and creating "unnecessary drama", said he signed off the payment.
He said: "I don't remember the email, but I have read them and thought a lot about them (in preparation for the trial). I didn't believe Clive was paying policemen, I still don't believe it.
"I think what I did was failed to address it properly with him. But I didn't (address it), I rubber-stamped it."
Mr Coulson denies two counts of conspiring with Mr Goodman, 56, of Surrey, and others to commit misconduct in a public office. Mr Coulson, 46, of Charing, Kent, is also charged with conspiring to hack phones with ex NotW boss Rebekah Brooks and former managing editor Stuart Kuttner.
All seven defendants deny all charges against them. The trial at the Old Bailey continues.