* "Enough is enough," says PM Valls
* Dozens hurt in Tuesday street battles
* French police stretched by several challenges (Adds CGT comments)
By Leigh Thomas and Brian Love
PARIS, June 15 (Reuters) - French leaders told the hardline CGT labour union on Wednesday it would be denied permission for further street rallies unless it rooted out troublemakers, a day after violent battles between masked youths and police during protest marches in Paris.
Prime Minister Manual Valls accused the CGT of doing little to rein in hundreds of rioters who ransacked shopfronts, tore up street paving and smashed the windows of a children's hospital during running battles on Tuesday. The police responded with teargas and water cannon and dozens were hurt on both sides.
President Francois Hollande told ministers street rallies that the CGT has been organising in protest against a labour law reform would not be permitted unless the union provided better security guarantees, a government spokesman said.
In a grim assessment of the broader climate in France, Valls said terrorists would inevitably strike again, that police were also working overtime to avert fan violence during a month-long European soccer championship and that the anti-reform protests had now degenerated into "unbearable violence".
He was speaking two days after a police officer and his wife were knifed to death by a Frenchman pledging allegiance to the Islamic State militant group.
"Enough is enough," Valls said. "I am calling on the CGT to hold no further demonstrations in Paris," he told public radio.
He accused the union of maintaining an "ambiguous" attitude towards some 700 ultra-violent youths.
The CGT hit back in a statement, saying it was up to the state to guarantee order. And in an interview with France 2 television, the CGT's general secretary, Philippe Martinez, said his organisation would call for new demonstrations on June 23 and June 28.
"It would be wise for the government to suspend the discussions over the bill in Parliament," he said.
Previous street rallies over the planned labour reform have been marked by violence, too, but Valls said Tuesday had reached a new peak. "These are people who want to take out a police officer, to kill a police officer," he said.
Police arrested more than 60. Many of the troublemakers were clad in black, with scarves and goggles to repel tear gas and helmets in case of baton charges. While they belonged to no single identifiable group, experts said they appeared to be mostly disillusioned youths, anarchists and anti-capitalists.
About 75,000 to 80,000 people took part in the Paris march in all, according to police estimates.
Valls reiterated that his government would not back down on a labour reform that will make hiring and firing easier and devolve the setting of work conditions to company level. He wants it adopted by July.
His government forced the bill through the lower house of parliament by decree to bypass rebels within the ruling Socialist Party ranks and it is now being debated in the Senate.
France has allowed the street marches so far despite the emergency rule imposed after Islamist militants killed 130 people in attacks in Paris last November.
Police are also striving to keep the peace during a soccer tournament that runs till July 10. They stepped in at the weekend to break up violence between Russian and English fans and are on alert for further trouble during upcoming matches.
Valls said this came on top of a permanent terror threat.
"More innocents will lose their lives," he said. "We need to tighten the net and give police and intelligence services all the means they need, but we will witness further attacks." (Additional reporting by Matthias Blamont, Emmanuel Jarry; Writing by Brian Love; Editing by Andrew Callus and Larry King)
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