This paper proposes a simple yet effective framework of soft cross-lingual syntax projection to transfer syntactic structures from source language to target language using monolingual treebanks and large-scale bilingual parallel text. Here, soft means that we only project reliable dependencies to compose high-quality target structures. The projected instances are then used as additional training data to improve the performance of supervised parsers. The major issues for this idea are 1) errors from the source-language parser and unsupervised word aligner; 2) intrinsic syntactic non-isomorphism between languages; 3) incomplete parse trees after projection. To handle the first two issues, we propose to use a probabilistic dependency parser trained on the target-language treebank, and prune out unlikely projected dependencies that have low marginal probabilities. To make use of the incomplete projected syntactic structures, we adopt a new learning technique based on ambiguous labelings. For a word that has no head words after projection, we enrich the projected structure with all other words as its candidate heads as long as the newly-added dependency does not cross any projected dependencies. In this way, the syntactic structure of a sentence becomes a parse forest (ambiguous labels) instead of a single parse tree. During training, the objective is to maximize the mixed likelihood of manually labeled instances and projected instances with ambiguous labelings. Experimental results on benchmark data show that our method significantly outperforms a strong baseline supervised parser and previous syntax projection methods. 1