Romania's interior ministry said 11 crew members of the ship which sank had been rescued.
The country's naval authority said the incident involved a fire on the ship Royad Mammadov.
Romanian authorities did not comment on the cause of the fire but Ukrainian President Volodymyr Zelenskiy said it had been caused by Russian drones.
He confirmed the death of one person in the incident but said a second person had yet to be accounted for.
"A horrific strike by two Russian drones on a civilian vessel in neutral waters carrying corn and owned by Türkiye. Azerbaijani and Indian citizens were on board," Zelenskiy wrote on social media platform X.
"As of now, unfortunately, we know that the ship's captain was killed. My condolences. The fate of one other person is still being established. Eleven crew members were rescued, one of them injured."
Russian authorities had no immediate comment.
An earlier daily bulletin from the Russian defence ministry said Russia had struck two cargo vessels.
Romanian media earlier reported that the incident had occurred near an oil platform operated by Romanian energy company OMV Petrom, which told Romanian website HotNews.ro that the platform was operating normally.
The ship, flying under the flag of St Kitts & Nevis, had been headed to Ravenna, Italy, and had last docked at the Ukrainian port of Izmail, according to website VesselFinder.com.
NATO member Romania shares a 614km land border with Ukraine.
Since the Ukraine-Russia war started four years ago, it has had Russian drones breaching its airspace.
Mines have also floated in the Black Sea across key trade and energy routes.
Indian Foreign Minister Subrahmanyam Jaishankar said on Monday that his country was pursuing talks with Russia and Ukraine for the safe passage of grain and fuel shipments from the Black Sea.
