In this paper we solve the embedding problem for partial transitive triple systems of order n and index λ, partial TTS (n, λ) s, showing that any partial TTS (n, λ) can be embedded in a TTS (v, λ) for all λ-admissible v[ges ]2n+1. This lower bound on v is the best possible. A simple proof is also given of a result of Colbourn and Harms which shows that every partial triple system of order n and index 2λ is the underlying triple system of a partial TTS (n, λ).