The remark was made after Biegun's meeting with South Korean representative Lee Do-hoon for Korean Peninsula peace and security affairs in Seoul commenced.
North Korea last week called South Korean President Moon Jae-in an "impudent guy rare to be found" for hoping to resume inter-Korean talks while continuing military drills with the US.
Last week, Trump played down the North's recent missile tests, saying they do not violate Kim's pledge to forego nuclear and long-range tests.
USA diplomat Stephen Biegun said, "We are prepared to engage (in a working-level dialogue with the DPRK) as soon as we hear from our counterparts in North Korea (DPRK)".
It added the drills are an "open disregard and wanton violation" of the joint statement adopted in the first US-North summit in Singapore a year ago, in which US President Donald Trump and North Korean leader Kim Jong Un agreed to pursue new US-North relations for "peace and prosperity".
Trump has voiced his discontent as well, not about North Korea's missile tests, but about the costs of the military drills with Seoul.
Lee, the South Korean nuclear envoy, told reporters that he had a productive dialogue with Biegun as they discussed how to rapidly resume talks between Pyongyang and Washington and make substantive progress in the negotiation for the denuclearization.
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A shop sign in Seoul displaying banned Japanese products as a trade dispute between the USA allies worsens. -South Korean drills ended and that Kim offered him "a small apology" over the weapons tests.
Despite their close economic interdependence, ties between the three have often been fraught over trade frictions, the role of the USA and lingering resentment over Japan's colonial legacy and World War II aggression.
South Korea is reinforcing safety inspections of food products imported from Japan, including the nuclear-hit region of Fukushima, amid rising concerns over radiation.
"It is possible that North Korea has achieved the miniaturisation of nuclear weapons and has developed nuclear warheads".
"I am fully committed to this important mission", Biegun said. Trump on Tuesday indicated he may nominate Deputy Secretary of State John Sullivan to the position.
"I believe it will start in late August or early September".
KCNA said Kim had overseen the launch of the unspecified "new weapon" and expressed "great satisfaction" over his military's "mysterious and fantastic success rates" in recent testing activity. During their third meeting at the Korean border village of Panmunjom on June 30, the two leaders agreed to restart the talks but there has been no public meeting between the countries.