With the 1948 renaming of the degree centigrade to degree Celsius also came a redefinition in terms of the triple point of water, the temperature at which solid, liquid, and gas coexist. So it no longer used the melting point of water, now approximately 0,00°C instead of exactly. The triple point is at 0,01°C by definition.

In 1954 the subsequent Conférence Générale des Poids et Mesures redefined the absolute temperature scale to use the triple point, and the effect of this was to remove the exact value of the boiling point too. The degree Kelvin (renamed in 1967 as the kelvin) was defined to be the fraction 1 / 273,16 of the triple point of water. As the degree Celsius is identical in size to the kelvin, and as the triple point is 0,01°C exact by definition, this means that the Celsius scale now starts with its zero at 273,15 K, which is approximately the melting point of water, and with the boiling point at approximately 100,00°C or 373,15 K.

The next CGPM in 1960 created the SI, and the degree Celsius was not part of it: it was considered an everyday unit rather than a scientific one, the only SI unit for temperature being the degree Kelvin. The degree Celsius was relegated to a class with the hour, tonne, bar and a few others of important non-SI units still used with the SI. However in recent years this was changed and the degree Celsius was readmitted as an SI unit proper.

This means it can take all the standard SI prefixes, and this has an odd effect on the symbol. Unlike all the other SI symbols, the degree sign is attached to its number: 10°C, not 10 °C. But the prefixed ones take the space as normal: ten millidegrees Celsius is symbolized 10 m°C

The Celsius scale is also known as the practical scale. The Celsius temperature is symbolized t and is related to the thermodynamic temperature T by the equation t = T - T0, where T0 is the Celsius starting-point 273,15 K.

SI definitions and history at the BIPM official site: www1.bpm.org
In particular, degree Celsius included among derived units: http://www1.bipm.org/en/si/derived_units/2-2-2.html
The unit was readopted by the SI apparently by a 1989 decision of the CIPM, not the full CGPM: http://www1.bipm.org/en/si/base_units/kelvin.html