This is a community wiki answer collecting some of the comments:
While it seems unlikely that Deligne's article has been translated, there are several alternative sources in English that may be of use, including:
Tate's article Number-theoretic background from volume 2 of the Corvalis proceedings. It provides a lot of introductory material on Weil groups (some of it taken from Deligne's paper).
Chapter 7 of Bushnell & Henniart's book The Local Langlands Conjecture for GL(2) (Springer-Verlag, 2006, MR2234120). It presents (with proofs) the material on Weil groups and their (Weil–Deligne) representations as well as proving the existence of the "local constants", thus covering a lot of what is in Deligne's article. It even has a proof of the ℓ-adic monodromy theorem previously only available in the appendix to Serre–Tate!
Tate's article Local constants, in Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 89–131. Academic Press, London, 1977.
[Anyone reading this should feel free to add more bibliographic information, further references, or additional commentary on the various suggestions.]
