Shabbos, Chapter 12, Mishnah 6. Before we begin, let us set out the general rule: a person who performed a melachah on Shabbos and then performed it a second time is liable to bring two chatas offerings - provided that he performed the two acts in two separate lapses of awareness, that is, two distinct instances of forgetting: he forgot that it was Shabbos and performed a melachah, then remembered that it was Shabbos, and then forgot again and performed the melachah once more. But if he performed the melachah twice within a single lapse, within one forgetting of Shabbos, he brings only one offering.
The novelty of our Mishnah - two half melachos in two lapses:
Our Mishnah deals with a case in which two complete melachos were not performed, but rather two halves of a melachah, each one in a different lapse of awareness. The Mishnah states: "hakosev shtei osiyos bishnei helamos" - a person who forgot that it was Shabbos and wrote only one letter, then remembered that it was Shabbos, and afterward forgot again and wrote a second letter. The question is whether the two letters combine with one another to make up the measure that carries liability, or whether they do not combine.
The Mishnah offers an additional example: "achas shacharis ve'achas bein ha'arbayim" - one letter written in the morning and one letter written toward evening. Rashi explains that writing in the morning and writing in the afternoon are separated from one another to such a degree that their law is like that of two letters written in two lapses of awareness, in two distinct instances of forgetting. Here too the Tannaim disagree.
The dispute between Rabban Gamliel and the Sages:
Rabban Gamliel holds him liable - in his view the two letters combine, because there is no such thing as awareness regarding half a measure. The awareness that it is Shabbos does not take effect with respect to half a measure, and it does not have the power to split the melachah into two. The fact that the person became aware in the middle of Shabbos makes no difference here, and therefore the two halves combine to create liability.
And the Sages exempt him - in their view there is awareness regarding half a measure. Just as awareness separates between two complete melachos performed in two lapses, so too it separates between the two halves of the melachah before us, and they cannot be joined together to create liability - and therefore he is exempt.
In summary: our Mishnah discusses two halves of a melachah - two letters - written in two lapses of awareness, whether in a sequence of forgetting, awareness and forgetting again, or one in the morning and one toward evening. According to Rabban Gamliel there is no awareness regarding half a measure, the letters combine, and the writer is liable; according to the Sages there is awareness regarding half a measure, the awareness separates between the halves, and the writer is exempt.