No. The Pope did not permit or condone artificial birth control.
There's a full critique of a number of statements from Light of the World, together with statements from the Vatican which provide a gloss, on the website of the Holy Family Monastery, although it appears they consider that the Pope has erred simply because his statements will be used to misrepresent his and the Church's teaching on contraception.
In Light of the World p117–119, we read this:
Question: “In Africa you stated that the Church’s traditional teaching has proven to be the only sure way to stop the spread of HIV. Critics, including critics from the Church’s own ranks, object that it is madness to forbid a high-risk population to use condoms.”
Answer from Benedict XVI: “The media coverage completely ignored the rest of the trip to Africa on account of a single statement… In my remarks I was not making a general statement about the condom issue, but merely said, and this is what caused such great offense, that we cannot solve the problem by distributing condoms. Much more needs to be done… There may be a basis in the case of some individuals, as perhaps when a male prostitute uses a condom, where this can be a first step in the direction of a moralization, a first assumption of responsibility, on the way toward recovering an awareness that not everything is allowed and that one cannot do whatever one wants. But it is not really the way to deal with the evil of HIV infection. That can really lie only in a humanization of sexuality.”
Question: “Are you saying, then, that the Catholic Church is actually not opposed in principle to the use of condoms?
Answer from Benedict XVI: “She of course does not regard it as a real or moral solution, but, in this or that case, there can be nonetheless, in the intention of reducing the risk of infection, a first step in a movement towards a different way, a more human way, of living sexuality.”
The Congregation for the Doctrine of the Faith stated on 19 December 2010 (my emphasis):
Some interpretations have presented the words of the Pope as a contradiction of the traditional moral teaching of the Church. This hypothesis has been welcomed by some as a positive change and lamented by others as a cause of concern — as if his statements represented a break with the doctrine concerning contraception and with the Church’s stance in the fight against AIDS. In reality, the words of the Pope — which specifically concern a gravely disordered type of human behaviour, namely prostitution (cf. Light of the World, pp. 117–119) — do not signify a change in Catholic moral teaching or in the pastoral practice of the Church... The idea that anyone could deduce from the words of Benedict XVI that it is somehow legitimate, in certain situations, to use condoms to avoid an unwanted pregnancy is completely arbitrary and is in no way justified either by his words or in his thought.
[L’Osservatore Romano 22-27 December 2010, p11]
TL;DR: Condoms are not to be used to prevent pregnancy, but they can be used to prevent the spread of disease. Someone recognising that this is necessary may be a first step towards the moral and responsible stance of abstinence.