In the statement issued by the Metropolitan Police on behalf of the 33-year-old, Ms Skripal revealed she “woke up over a week ago” after being exposed to a nerve agent in the Wiltshire city.
She said: “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.”
In the statement, Ms Skripal made no mention of the condition of her father Sergei.
The Skripals have been treated at Salisbury District Hospital since being found collapsed on a public bench on 4 March.
The UK Government blames Russia for the use of military-grade novichok in the attempted murder of ex-Russian spy Mr Skripal and his daughter. Moscow has denied the accusations.
The Russian Ambassador to the UK, Alexander Yakovenko, told journalists on Thursday he was “really happy” about the news of Ms Skripal’s condition and expressed hope her father would also recover.
He added he was “quite sure” Ms Skripal would return to her job and apartment in Moscow, but it would be the “choice” of Mr Skripal whether or not he returns to Russia.
The 66-year-old was sentenced to 13 years in a Russian jail in 2006 after being accused of acting as a double agent for Britain.
The former intelligence officer was part of a spy swap with the US in 2010, after which he flew to the UK and settled in Salisbury.
“From the point of the Russian state, Russia has no problems with him,” Mr Yakovenko said.
“Because, you know, he was sentenced, he spent several years in a prison and then he’s like any other citizens, he’s clear, that’s it.
“So, for us, they [the Skripals] are just Russian citizens.”
The Foreign Office revealed Ms Skripal has so far rejected the Russian embassy’s offer of consular help.
A spokesperson said: “We are pleased that Yulia’s condition has improved and pay tribute to the medical staff who have been treating her and Mr Skripal since they were attacked.
“We have conveyed to Ms Skripal the Russian embassy’s offer of consular assistance.
“Ms Skripal is now able to choose if and when to take up this offer, but to date she has not done so.”
Ms Skripal’s statement came after Russian state TV aired what it said was a phone conversation between Ms Skripal and her cousin Viktoria Skripal, which it said it obtained from the latter.
Ms Skripal was quoted as having told the cousin: “Everything is fine, everything is fixable, everyone is getting better.”
Asked about her father, Ms Skripal was said to have added: “Everything is fine, he is resting, sleeping. Everyone’s health is fine.”
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Ms Skripal was also reported to have claimed she would soon be discharging herself from hospital.
On Thursday evening, a meeting of the UN Security Council in New York will discuss the Salisbury poisoning amid a deepening diplomatic rift between Moscow and London.