Yulia Skripal, the daughter of a former spy poisoned in Britain, has said "my strength is growing daily" and thanked the people of Salisbury who "came to my aid."
In a statement issued through the Met Police, the 33-year-old also asked for her privacy to be respected "during the period of my convalescence."
It made no reference to the condition of her father, Sergei.
The statement reads: "I woke up over a week ago now and am glad to say my strength is growing daily. I am grateful for the interest in me and for the many messages of goodwill that I have received.
"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 statement came shortly after Russian state TV broadcast what was alleged to be a call between Yulia, 33, and her cousin.
Viktoria Skripal said they had spoken by telephone and a recording of the alleged call has been given to the Rossiya-1 TV channel.
Its authenticity has not been verified. A transcript of the alleged call has been published by the BBC.
In it, the woman claiming to be Yulia said her father Sergei "is all right."
"Everyone is recovering, everyone survived... No irreparable harm was done. That's all, I will soon be discharged from the hospital," she is alleged to have said.
Transcript of alleged 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.
This transcript was published in English by the BBC:
Alleged Yulia: Hello. Do you hear me?
Viktoria: Yes, I hear you.
Alleged Yulia: It is Yulia Skripal.
Viktoria: Oh, Yulka [diminutive of Yulia] it is you! I recognise from your voice that it is you but cannot understand. So, they gave you a telephone, didn't they?
Alleged Yulia: Yes, yes.
Viktoria: Thanks God! Yulyash [diminutive of Yulia], is everything okay with you?
Alleged Yulia: Everything is ok, everything is fine.
Viktoria: Look, if tomorrow I get a (British) visa, I will come to you on Monday.
Alleged Yulia: Vika, no-one will give you the visa.
Viktoria: Well I thought so too. Oh well.
Alleged Yulia: Most likely.
Viktoria: If they give it, I need you to tell me whether I can visit you or not, tell me that I can.
Alleged Yulia: I think no, there is such a situation now, we'll sort it out later.
Viktoria: I know it, I know it all.
Alleged Yulia: Later, we will get it sorted later, everything's fine, we'll see later.
Viktoria: Is it your phone?
Alleged Yulia: It is a temporary phone. Everything is 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 (he and her father) is recovering and is alive.
Viktoria: Clear! Is everything ok with your father?
Alleged Yulia: Everything is ok. He is resting now, having a sleep. Everyone's health is fine, there are no irreparable things. I will be discharged soon. Everything is ok.
Viktoria: Kisses, my bunny.
Alleged Yulia: Bye.
On Wednesday evening, Viktoria Skripal told state TV that she had applied for a British visa in order to visit the hospital, but that she was unsure whether British authorities would allow her to see her cousin or her uncle Sergei, who were poisoned on March 4 and remain in hospital.
She told the Rossiya-1 60 Minutes TV show, which has frequently cast the nerve agent attack as part of an elaborate British plot to smear Russia: "I currently have one objective: to fly there and take away Yulia, at least Yulia." She said the situation is much more complicated for Sergei Skripal, 66.
She said she was in regular contact with Russia's ambassador to London, Alexander Yakovenko, and that he had helped her obtain a Russian passport to travel.
At a press conference on Thursday, the Russian ambassador to the UK quoted the results of a Twitter poll as evidence that "most of the British public" believe the results of investigations will be covered up.
Viktoria Skripa previously said there was a slim chance the pair would survive the attack, saying the prognosis "really isn't good."
"Out of 99% I have maybe 1% of hope. Whatever it was has given them a very small chance of survival. But they're going to be invalids for the rest of their lives," Viktoria Skripal told the BBC.
Britain has blamed Moscow for the chemical attack, something Russia denies.
At the press conference on Thursday the Russian ambassador said Foreign Minister Boris Johnson's claim that Moscow maintains a stockpile of the lethal nerve agent Novichok "likely for assassinations" is "not true... not supported by any evidence... unacceptable."
The case has plunged East-West relations into their worst crisis since the Cold War, with Britain and its allies expelling around 130 Russian diplomats and Moscow responding with its own expulsions.