* Victoria reports 331 new COVID-19 cases, 19 deaths
* NSW reports biggest one-day rise in cases since April
(Adds national daily COVID-19 tally, details on borders to stay closed)
By Colin Packham
SYDNEY, Aug 11 (Reuters) - Australia's second-most populous state reported only a small rise in new COVID-19 infections on Tuesday, boosting hopes that case numbers are stabilising after a second wave forced authorities to put the city of Melbourne back into lockdown.
Victoria state, which currently accounts for nearly all of Australia's new cases, detected 331 COVID-19 infections and 19 deaths in the past 24 hours, up from 322 infections and the same number of fatalities a day earlier, health officials said.
Daily infections in Victoria peaked at 725 on Aug. 5 and have been trending lower in recent days, following the imposition of a hard lockdown in Melbourne on July 19.
While the lockdown has caused significant economic harm, authorities said the restrictions that will run until September are bearing fruit.
"We continue to see numbers coming down. Exactly how long that takes and to what the lowest number is we can get to, only time will tell," Victorian Premier Daniel Andrews told reporters in Melbourne.
New South Wales state reported 22 new cases, the biggest one-day rise since April 17.
About a third of those cases were linked to an outbreak at a school in Sydney, the source of which remained unknown, NSW health officials said.
"It's not always possible to determine the source and that's what worries us most," NSW Health Minister Brad Hazzard said.
No other cases were reported elsewhere in Australia. The country has confirmed almost 22,000 infections and 332 deaths from COVID-19, far fewer than many other developed nations.
Still, Australia was recording less than 20 new cases a day as recently as June, when lockdowns were eased.
Quarantine breaches in Victoria are believed to be the source of the state's second wave.
Victorian Premier Andrews appeared as a witness on Tuesday at an inquiry into the state's handling of the outbreak.
He said a lack of resources forced the state government to employ private security in hotels used to quarantine people returning to Australia from overseas.
Victoria is cut off from the rest of Australia's states and territories, which have closed their borders to residents of places deemed to be virus hotspots.
The remote Northern Territory said it would keep its borders closed for at least 18 months.