You have to add the sourcing inside the .bashrc.
This works:
FROM ubuntu:16.04
# Update default packages
RUN apt-get update
# Get Ubuntu packages
RUN apt-get install -y \
build-essential \
curl
# Update new packages
RUN apt-get update
# Get Rust
RUN curl https://sh.rustup.rs -sSf | bash -s -- -y
RUN echo 'source $HOME/.cargo/env' >> $HOME/.bashrc
EDIT
Instead of
RUN echo 'source $HOME/.cargo/env' >> $HOME/.bashrc
you can use
ENV PATH="/root/.cargo/bin:${PATH}"
which is a less bash-only solution