In 2013, Cambridge University researcher Aleksandr Kogan is alleged to have collected data from 50 million users using a quiz app before passing the information to election consultancy Cambridge Analytica (CA) in 2014.
Mr Zuckerberg said he was "really sorry" for the "major breach of trust" and pledged to work to prevent data from being misused in future.
After thanking users for "believing in this community", Zuckerberg vowed to "do better for you".
Claims that the data acquired by CA may have been used as part of Donald Trump's 2016 presidential campaign sparked an worldwide backlash against Facebook when they emerged last week.
The plain black text apology on a white background, with only a tiny Facebook logo, appeared in Sunday publications including The Observer - one of the newspapers whose reporting on the issue has sent Facebook's share price tumbling.
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The scandal centers around British data company Cambridge Analytica, which has ties to Donald Trump's 2016 presidential campaign. "We're now taking steps to ensure this doesn't happen again", Zuckerberg writes, adding that the company has already "stopped apps like this from getting so much information".
A new Reuters-Ipsos poll in the USA released Sunday showed that 41 percent of Americans trust Facebook to obey laws that protect their personal information, compared to 66 percent of trust in Amazon; 62 percent in Google; 60 percent in Microsoft and 47 percent in Yahoo.
Information Technology Minister Ravi Shankar Prasad on Wednesday warned against any abuse of social media in elections, following reports that a British consultancy improperly accessed information on millions of Facebook users to target USA voters.India is due to hold a national election in 2019 and several states will elect new assemblies this year and next.
Zuckerberg said Facebook, with 2.2 billion users worldwide, is also investigating "every single app that had access to large amounts of data before we fixed this". "And when we find them, we will ban them and tell everyone affected".
"We have a basic responsibility to protect people's data", Zuckerberg said in an interview with CNN, breaking a public silence since the scandal erupted at the weekend. I promise to do better for you.