One Western editor promised 'to make me a millionaire if I would say that I had French sex with Beria', she says. And Russian journalists 'twisted' Beria's words to her to make him appear more brutal than he was 'for political ends'.
The truth, she says, is rather different. Although she says she was an unwilling lover of the man who sent countless numbers of innocent people to their deaths during Stalin's terror, 'he was not a monster to me. He was gallant and affectionate. Why should I sin by speaking ill of the dead?'
Ms Alexeva is lucky to have survived, for most of Beria's victims never came out alive from his mansion on Moscow's Kachalova Street, which now houses the Tunisian embassy.
As head of Soviet security, Beria was responsible for deporting the people of the Baltic states to Siberia. He not only used to interrogate suspects in this house but also, it is believed, to lure young women in, then rape and murder them. Recently workmen digging up the street outside found several skeletons covered in lime, almost certainly the remains of some of Beria's 'guests'.
Ms Alexeva, who sits in her carpet-strewn apartment wearing a scarlet silk dressing-gown and dark glasses, coquettishly pretending that she does not want to be photographed, was a singer in the choir of the NKVD (later called the KGB) when she caught Beria's eye at a concert in the late 1940s. When the secret police chief sent aides to make inquiries about her, it came to the attention of a colleague of her husband, who warned the couple. So, one day when a black limousine pulled up beside her and the driver said, 'A certain person likes you,' she had the presence of mind not to get in but to say that she was waiting for her husband, army Colonel Ivan Rebrov. The driver excused himself and the car glided off. 'God saved me that time,' she says.
Soon afterwards, Rebrov was arrested. Ms Alexeva is still not sure whether Beria deliberately removed this obstacle to the satisfaction of his lust or whether her husband just disappeared, like millions of other people rounded up in night raids and shot or sent to slower deaths in labour camps. Ms Alexeva fled to Kaliningrad, on the Baltic coast, where she felt Beria could not reach her.
After a year she thought it would be safe to return. Back in Moscow she rejoined the choir and fell in love with a naval officer, who was to become her next husband. But Beria had not forgotten her. He sent bodyguards to bring her to him - and this time she could not refuse.
Ms Alexeva was shown into an elegant dining room where a table was laid with 'zakuski (hors d'oeuvres) of all kinds and delicious wines from the cellars of Tsar Nicholas II. It was the kind of food ordinary people never saw'. After dinner, Beria took her to bed. 'I was trembling all over. I did not want this to happen,' she said, although she would not use the word 'rape'.
'He was tender, he stroked my hair and kissed me. Maybe he was cruel to others, but he liked me.' Beria did not do 'anything perverted', and there was definitely no oral sex. 'I was not that sort of girl. I was from a good family. I was a homely sort of person.'
Beria let her go next morning, giving her a large bunch of red roses. Every two or three days for the next 18 months, the car was sent for her, and she spent the night with the secret policeman while her new husband Dmitry chafed at home. 'I was not Beria's mistress, I was his victim. I had fallen into his net.' The young woman was confined to the bedroom, the dining room and the bathroom, which had a marble table on which Beria lay for massages. The other rooms were closed to her and she never saw his other victims. 'I didn't know about his bloody business then,' she says. 'No one did really. We could only guess.'
Beria, whom she described as witty, cunning and a great personality, took her to Marshal Zhukov's dacha and to meet Stalin at a reception. He offered to help her singing career and to set her up with an apartment and her dream, a grand piano. She says she refused to accept any privileges or gifts from him and concentrated on devising ways of getting out of his clutches. She trusted one of Beria's bodyguards, Raphael Semoyonovich, and begged him to help. He found another woman, also from the choir, to replace her. Amazingly, Beria agreed to let her go. At their last meeting he told her: 'I know you are cold to me so I must accept my fate soberly.' Ms Alexeva says: 'I thanked him profusely, for I was a good actress.'
She puts her escape down to patience and her ability to keep a cool head. She thinks Beria may have pitied her because she reminded him of his own wife, a Georgian woman also called Nina. What really saved her, probably, was Beria's death, soon after their affair ended in 1953. Stalin was gone and the men who succeeded him, fearing Beria would continue the terror, had him shot on the absurd charge that he was a British spy.
Ms Alexeva went on to work in television. Now she is a grandmother, living in a tower block on a pension of 16,000 roubles ( pounds 11) a month. She will not admit it, but she clearly loves the attention she is receiving as a result of the revelations about her affair with one of history's most notorious mass- murderers.