Ban called on the government and the opposition to take measures to ease the delivery of relief supplies, notably medicine, to 9.3 million Syrians in need.
"One month after the passing of Security Council resolution 2139, humanitarian access in Syria remains extremely challenging for humanitarian organizations," Ban said.
"Delivering lifesaving items, in particular medicines, remains difficult. And the assistance reaching people continues to fall far short of what is required to cover even their basic needs," he said.
The resolution, which Russia supported after negotiations, provides no sanctions against those who fail to respect it.
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While it leaves open the door to sanctions against violators, based on Ban's report, that would require another decision by the Security Council.
Diplomats expect that Russia would block any sanctions against the Syrian regime, as it has done three times since the start of the Syrian crisis in March 2011.
"I strongly condemn the continued heavy shelling, including the use of barrel bombs by the Syrian government forces in residential neighborhoods, as well as the terror acts in Syria by extremist groups who are attempting to impose radical ideologies in some parts of the country," Ban said.
He appealed to states and other actors in the Syrian civil war to end their support for the violence, and to use their influence to promote a political solution the conflict, now in its fourth year.