Given a smooth hyperquadric Y in P^4, we consider its blowup X along a smooth irreducible curve C contained in Y. We study the question, when X is a weak Fano threefold, that is, when it has a nef and big anticanonical divisor. We are able to give a complete classification of such threefolds only depending on some geometric properties of C, particularly its genus and degree. We introduce the main proof ideas which include the analysis of the linear system given by the anticanonical divisor of X, as well as studying curves contained in a smooth K3 surface of degree 6.