The De Broglie wavelength is proportional to 1/momentum, so when the magnitude of momentum (which for slow-moving particles is mass times speed) small, the De Broglie wavelength gets large. When momentum goes to zero, the relations say the wavelength is infinite. This also matches with the uncertainty principle, which says that if you know that the particle's momentum is exactly zero, you have no idea where in the universe it is.
In practice, you're never going to be able to get a zero-momentum particle, since no matter how much you try to slow things down, there is always some tiny quantum mechanical wiggling about going on. The De Broglie wavelength is also mostly a handy rule of thumb for determining the size of things. To really make predictions you generally need to use a more thorough version of quantum theory.
Edit: It might clarify the question if I explain what the De Broglie wavelength is. Quantum mechanically, matter can be described in terms of particles or waves. The De Broglie wavelength is the distance between peaks of the quantum wave describing matter. It's also roughly the answer you get if you try to measure the size of a quantum mechanical particle.