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We present {\tt deepSIP} (deep learning of Supernova Ia Parameters), a software package for measuring the phase and -- for the first time using deep learning -- the light-curve shape of a Type Ia supernova (SN~Ia) from an optical spectrum. At its core, {\tt deepSIP} consists of three convolutional neural networks trained on a substantial fraction of all publicly-available low-redshift SN~Ia optical spectra, onto which we have carefully coupled photometrically-derived quantities. We describe the accumulation of our spectroscopic and photometric datasets, the cuts taken to ensure quality, and our standardised technique for fitting light curves. These considerations yield a compilation of 2754 spectra with photometrically characterised phases and light-curve shapes. Though such a sample is significant in the SN community, it is small by deep-learning standards where networks routinely have millions or even billions of free parameters. We therefore introduce a data-augmentation strategy that meaningfully increases the size of the subset we allocate for training while prioritising model robustness and telescope agnosticism. We demonstrate the effectiveness of our models by deploying them on a sample unseen during training and hyperparameter selection, finding that Model~I identifies spectra that have a phase between $-10$ and 18\,d and light-curve shape, parameterised by $Δm_{15}$, between 0.85 and 1.55\,mag with an accuracy of 94.6\%. For those spectra that do fall within the aforementioned region in phase--$Δm_{15}$ space, Model~II predicts phases with a root-mean-square error (RMSE) of 1.00\,d and Model~III predicts $Δm_{15}$ values with an RMSE of 0.068\,mag.