UK police have issued a statement on behalf of Russian spy Sergei Skripal's daughter, the first since the pair were poisoned by a nerve agent in March.
The statement quoted Yulia Skripal as saying her "strength is growing daily".
Ms Skripal, 33, and her father are in hospital in Salisbury, southern England, where the attack took place.
The UK Foreign Office said Ms Skripal, who is a Russian citizen, has not yet taken up Russia's offer of consular assistance.
The Russian embassy in London said last week that it was insisting on its right to see her after it emerged that she was conscious and talking.
Earlier on Thursday, Russian TV aired a recording of an alleged phone conversation between Ms Skripal and her cousin.
However, doubts have been cast on how authentic the recording is; the presenters of the programme said they had been unable to verify it and themselves had doubts about it.
Mr Skripal, 66, remains critically ill but stable.
The UK government has accused Russia of being behind the 4 March attack, but Russia's ambassador in the UK said Moscow had no nerve agent stockpile.
The incident has sparked an international diplomatic crisis.
In the latest developments, Russia has accused the UK of inventing a "fake story" and "playing with fire" over the poisoning.
Moscow's UN ambassador Vasily Nebenzya told a meeting of the UN Security Council: "We have told our British colleagues that you are playing with fire and you will be sorry."
Moscow called the special meeting of the Security Council in New York to discuss the attack, saying Britain had "legitimate questions" to answer.
Mr Nebenzya called the accusations "horrific and unsubstantiated" and accused the UK of waging a "propaganda war" against Russia.
'Respect my privacy'
In the statement released by the police, Ms Skripal said she was grateful for the many messages of goodwill she had received.
"I woke up over a week ago now and am glad to say my strength is growing daily," the statement said.
"I have many people to thank for my recovery and would especially like to mention the people of Salisbury that came to my aid when my father and I were incapacitated.
"Further than that, I would like to thank the staff at Salisbury District Hospital for their care and professionalism.
"I am sure you appreciate that the entire episode is somewhat disorientating, and I hope that you'll respect my privacy and that of my family during the period of my convalescence."
The Russian TV recording is purported to be between Ms Skripal and her cousin Viktoria, who lives in Russia.
The woman described as Yulia says: "Everything is ok. He [her father] is resting now, having a sleep. Everyone's health is fine, there's nothing that can't be put right. I will be discharged soon. Everything is ok."
Viktoria Skripal said she hoped to travel to the UK to visit her relatives, if she could get a visa, but the voice said to be that of Yulia tells her that no one will give her one.
Correspondents say that what Yulia says in the recording about her father "resting" does not chime with what we know of the Skripals' health, and the conversation about the visa is strange.
The Foreign Office said its Moscow embassy was expected to give Viktoria a visa, possibly on Thursday, and that she would be given full consular help in the UK.
Russian news agency Interfax also quoted the cousin as saying in an interview that they had spoken and that Yulia had told her that all was well.
Viktoria Skripal expressed surprise that the phone call had come a day after she had told media that she was not allowed to speak to Yulia, Interfax reported.
"Do you believe in coincidences? I don't," it quoted her as saying.
Viktoria is said to be one of the pair's closest living relatives after a series of family deaths.
The poisoning, which the UK blames on Russia, has led to worsening relations between Moscow and many Western countries, with more than 150 diplomats expelled on each side.
Speaking shortly after the alleged conversation was released, Russian ambassador Alexander Yakovenko denied statements by UK officials that Russia did not respond to the poisoning allegations immediately.
He read out a letter which he said Russia had sent denying involvement, adding that claims that Russia had a nerve agent stockpile were not true.