I post this with some hesitation, since Taña has already cited this very phrase, but she has indicated that she prefers to leave her post as a reference. However, I am sure it's correct, and I feel that (if you agree) it ought to be put in the glossary, not least because every existing reference there to "título de abogado" equates it to a law degree, and that is not the case here, even if it is in other cases.

(By the way, it will be "practice law" in American English.)

Of course "título" often means a university degree, and "titularse" often means to graduate. But a "título de abogado" in Chile is a professional licence (license) to practise law, not a degree. The appropriate law degree, which you have to have before being granted the "título de abogado", is Licenciado en Ciencias Jurídicas (commonly known as Licenciado en Derecho), obtained from a university. To get the título you have to have one of these, and also fulfil certain other prerequisities: minimum age 20, no criminal record, certificate of good conduct, six months' probationary practice, Chilean nationality, or law studies in Chile if not Chilean.

The "título", as I say, is a professional licence. It would not be incorrect to say that this person was "granted the title of attorney/barrister", depending on whether the target is UK/Commonwealth or US; but "admission to practice law" is the standard expression for this:
http://en.wikipedia.org/wiki/Admission_to_practice_law

The "Corte Suprema" part is actually a bit of a tautology, since the "título de abogado" in Chile is always granted by the Supreme Court, in public plenary session. Here's a statement on the subject from the FAQ on the Chilean Poder Judicial website:

"En el ordenamiento jurídico chileno, los abogados son conceptualizados como las personas revestidas por la autoridad competente, de la facultad de defender ante los Tribunales de Justicia los derechos de las partes litigantes, de conformidad a lo dispuesto en el artículo 520 de nuestro Código Orgánico de Tribunales.
El título de abogado es otorgado por la Corte Suprema, en audiencia pública y reunida en tribunal pleno, y previa comprobación y declaración, de que el postulante reúne los requisitos establecidos por la ley, que son los siguientes:"
http://www.pjud.cl/PDF/ATUsuarios/PreguntasFrecuentes/requis...

It goes on to set out the requisites (including the licenciatura degree) as summarised above.

Here's someone else who includes the Corte Suprema part in his CV:

"Grados y títulos académicos
Licenciatura en Derecho, Pontificia Universidad Católica de Chile. 1994
Título de Abogado otorgado por la Corte Suprema de Justicia. 1998"
http://www.filosofiacristiana.cl/miembros/alvarado.html