NatWest said "we deeply regret" failing to "adequately monitor and therefore prevent money laundering by one of our customers between 2012 and 2016".
The state-backed bank, formerly Royal Bank of Scotland, is the first British lender to admit such an offence.
Lawyers told a court hearing that NatWest faces a large fine.
The case was brought by the Financial Conduct Authority (FCA) which alleged the bank failed to monitor suspect activity by a client that deposited about £365m in its accounts over five years, of which £264m was in cash.
The criminal action, first announced by the FCA in March, was the first against a bank under a 2007 money laundering law.
The FCA said NatWest failed to adhere to the requirements of anti-money laundering legislation in relation to Fowler Oldfield Ltd's account between 7 November 2013 and 23 June 2016.
Fowler Oldfield was a century-old jeweller based in Bradford, and was shut down following a police raid in 2016.
FCA prosecutor Clare Montgomery QC told Westminster magistrates that when Fowler Oldfield was taken on as a client by NatWest, its predicted turnover was said to be £15m per annum.
However, it deposited £365m over the space of almost five years.
She said: "It was agreed that the bank would not handle cash deposits. However, it deposited £365m, with around £264m in cash."
She said that at its height, Fowler Oldfield deposited up to £1.8m a day.
The court was told the "likely sentence is a very large fine".
NatWest remains 55% taxpayer-owned after receiving a £45bn bailout at the height of the 2008 financial crisis.
Chief executive Alison Rose said: "NatWest has a vital part to play in detecting and preventing financial crime and we take extremely seriously our responsibility to prevent money laundering by third parties.
"In the years since this case, we have invested significant resources and continue to enhance our efforts to effectively combat financial crime."
Jonathan Fisher, a senior lawyer at Bright Line Law specialising in money laundering cases, said the FCA had claimed "a big scalp".
"The message sent to financial institutions is clear. If there are failings in your money laundering systems, criminal prosecution may follow. NatWest was sensible to take the hit, and it must now move on."
A gang known as 400 Mawozo has been identified as behind the kidnapping of the group of 17 Missionaries in Haiti. The kidnappers...