The message, an invitation to an event at which attendees would be given a "Facebook phone number," was a phishing attack, designed to capture information from recipients.
Neither Facebook nor Breyer immediately responded to requests for comment.
Facebook director of communications Larry Yu told Thomson Reuters venture capital news site PE Hub that Breyer's account appeared to have been compromised and that the problem had been resolved.
A partner at Accel Partners, Breyer is one of four Facebook board members, the others being CEO Mark Zuckerberg, entrepreneur Marc Andreessen, and venture capitalist Peter Thiel.
The incident underscores the risk of supplying Facebook with data that might be better kept private.
Facebook's appeal to cybercriminals arises from the high level of trust that users extend to Facebook messages, which are generally presumed to come from friends.
Compromising someone's Facebook account also provides immediate access to a pool of new potential victims: the friends of the person whose account has been hacked.
Earlier this month, Versign's iDefense reported that its researchers had been monitoring a hacker in February who had been trying to sell login and password data for 1.5 million Facebook accounts.
Facebook told The New York Times that the hacker did not have the data that he claimed to have.
While it's not uncommon for cybercriminals to try to cheat each other, login data associated with online services from Facebook to Internet banks is regularly bought and sold on Web forums known as carding sites.
Update: After this story was filed, a Facebook spokesperson responded with the following statement: "We take security very seriously and have devoted significant resources to helping our users protect their accounts. We've developed complex automated systems that detect and flag Facebook accounts that are likely to be compromised (based on anomalous activity like lots of messages sent in a short period of time, or messages with links that are known to be bad). Because Facebook is a closed system, we have a tremendous advantage over e-mail. That is, once we detect a phony message, we can delete that message in all inboxes across the site. We also block malicious links from being shared and work with third parties to get phishing and malware sites added to browser blacklists or taken down completely. Users whose accounts have been compromised are put through a remediation process where they must take steps to re-secure their account and learn security best practices. This is what happened with Mr. Breyer's account."