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 says 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.
The United Nations Security Council has been discussing the poisoning in a meeting called by Russia.
Ambassador Vassily Nebenzia described the British accusations against Russia as horrific and unsubstantiated.
'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," it went on.
"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 says she hopes to travel to the UK to visit her relatives, if she can 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 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.
Transcript of recorded conversation
The hosts of the 60 Minute show on state-owned Rossiya 1 - Yevgeny Popov and Olga Skabeyeva - said they were unable to confirm the authenticity of the phone call.
"Yulia": Hello. Can you hear me?
Viktoria: Yes, I can hear you.
"Yulia": It is Yulia Skripal.
Viktoria: Oh, Yulka [diminutive of Yulia] it's you! I can tell it's you from your voice but I don't understand. So, they gave you a telephone, did they?
"Yulia": Yes, yes.
Viktoria: Thanks God! Yulyasha [diminutive of Yulia], is everything okay?
"Yulia": Everything's ok, everything's fine.
Viktoria: Look, if tomorrow I get a [British] visa, I'll come to you on Monday.
"Yulia": Vika, no-one will give you a visa.
Viktoria: Well I thought so too. Oh well.
"Yulia": Most likely.
Viktoria: If they do, I need you to tell me whether I can visit you or not, tell me that I can.
"Yulia": I don't think so, that's the situation at the moment, we'll sort it out later.
Viktoria: I know, I know.
"Yulia": Later, we'll get it sorted later, everything's fine, we'll see later.
Viktoria: Is that your phone?
"Yulia": It's a temporary phone. Everything's fine, but we'll see how it goes, we'll decide later. You know what the situation is here. Everything is fine, everything is solvable, everyone is recovering and is alive.
Viktoria: Understood. Is everything ok with your dad?
"Yulia": Everything's ok. He's resting now, having a sleep. Everyone's health is fine, there's nothing that can't be put right. I'll be discharged soon. Everything is ok.
Viktoria: Kisses, babes.
The recording was reportedly made on the morning of 5 April
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.