Given the low density of gas, the speed of sound would be a direct function of the temperature of the gas ie the speed of the molecules/atoms. Since this varies from about 2.7K to millions of degrees near some stars, the speed of sound can change quite a bit.

From the ideal gas law, we know: $$ v_textrm{sound} = sqrt{frac{gamma k_textrm{B} T}{m}} $$ Assuming that interstellar space is heated uniformly by the CMB, it will have a temperature of $2.73 mathrm{K}$. We know that most of this medium comprises protons and neutral hydrogen atoms at a density of about 1 atom/cm⁻³. This means that $gamma = 5/3$, and $m = 1.66times 10^{-27} mathrm{kg}$, giving a value for $v_textrm{sound}$ of $192 mathrm{m s^{-1}}$.

However, this is not propagated efficiently in a vacuum. In the extremely high vacuum of outer space, the mean free path is millions of kilometres, so any particle lucky enough* to be in contact with the sound-producing object would have to travel light-seconds before being able to impart that information in a secondary collision.

*Which for the density given, would only be about 50 hydrogen atoms if you clapped your hands – very low sound power!

-Edit- As has quite rightly been pointed out in the comments, the interstellar medium is not that cold. At the moment, our solar system is moving through a cloud of gas at approximately 6000 K. At this temperature, the speed of sound would be approximately $9000 mathrm{m s^{-1}}$.

See Kyle's answer for a table of values for $v_textrm{sound}$ that can be found in different environments in space, or pela's for information on how early universe sound waves became responsible for modern-day large scale structure.

You need to consider that space is filled with a tenuous plasma, which behaves slightly differently to an ideal gas. First, the electrons will carry sound at a different rate to the heavier protons, but also, the electrons and protons are coupled via the electric field. See: Speed (of sound) in plasma

The speed of sound in the solar wind is estimated at around 58 km/s, based on the equation in the answer given by Kieran Hunt. However, the temperature of the solar wind is more like $T = 1.2 times 10^5K$ (ref)

Just want to bring up that most answers seem to be taking "space" to be a nice uniform medium. However, even within our own galaxy, conditions vary wildly. Here are the most common environments in the Milky Way:

• Molecular Clouds, $rhosim 10^4,{rm atom}/{rm cm}^3$, $Tsim 10,{rm K}$
• Cold Neutral Medium, $rhosim 20,{rm atom}/{rm cm}^3$, $Tsim 100,{rm K}$
• Warm Neutral Medium, $rhosim 0.5,{rm atom}/{rm cm}^3$, $Tsim 10^4,{rm K}$
• Warm Ionized Medium, $rhosim 0.5,{rm atom}/{rm cm}^3$, $Tsim 8000,{rm K}$
• HII Region, $rhosim 1000,{rm atom}/{rm cm}^3$, $Tsim 8000,{rm K}$
• Hot Ionized Medium, $rhosim 10^{-3},{rm atom}/{rm cm}^3$, $Tsim ;{>}10^6,{rm K}$

The sound speed is proportional to $sqrt{T}$. Given that the temperature varies over about 7 orders of magnitude (maximum at about $10^7,{rm K}$, minimum at about $3,{rm K}$), the sound speed varies by at least a factor of $1000$. The sound speed in a warm region is on the order of $10,{rm km}/{rm s}$.

Trivia: the sound speed plays a crucial role in many astrophysical processes. This speed defines the time it takes for a pressure wave to propagate a given distance. One place this is a key time scale is in gravitational collapse. If the sound crossing time for a gas cloud exceeds the gravitational free fall time (time for a gravity-driven disturbance to propagate), pressure is unable to resist gravitational collapse and the cloud is headed toward the creation of a more compact object (denser cloud, or if conditions are right, a star).

More trivia: space is a very poor carrier (non carrier) of high frequency sounds because the highest frequency pressure wave that can be transmitted has a wavelength of about the mean free path (MFP) of gas particles. The MFP in space is large, so the frequency limit is low.

I know this question is technically already answered, but there were several things missing from the answers that I thought should be mentioned (I am writing a review paper comparing different regions of space so I had these numbers at hand already as well).

The speed of sound in space has multiple meanings because space is not a vacuum (though the number density of Earth's magnetosphere can be ~6-12 orders of magnitude more tenuous than the best vacuums produced in labs), it is full of ionized particles, neutral and charged dust.

In the interplanetary medium or IPM, there are five relevant speeds that can all be considered a type of sound in a way, because each is related to the speed of information transfer in the medium.

Classical idea of sound speed

When one discusses the speed of sound, one is generally referring to the common form of $C_{s}^{2} = partial P/partial rho$, where $P$ is the thermal pressure and $rho$ is the mass density. In a plasma, this takes the slightly altered form of: $$ C_{s}^{2} = frac{ k_{B} left( Z_{i} gamma_{e} T_{e} + gamma_{i} T_{i} right) }{ m_{i} + m_{e} } $$ where $k_{B}$ is Boltzmann's constant, $Z_{s}$ is the charge state of species $s$, $gamma_{s}$ is the adiabatic or polytrope index of species $s$, $m_{i}$ is the mass of species $s$, and $T_{s}$ is the average temperature of species $s$. In a tenuous plasma, like that found in the IPM, it is often assumed that $gamma_{e}$ = 1 (i.e., isothermal) and $gamma_{i}$ = 2 or 3, or that $gamma_{e}$ = 1 and $T_{e} gg T_{i}$. The above form of the sound speed is known as the ion-acoustic sound speed because it is the phase speed at which linear ion-acoustic waves propagate. Thus, $C_{s}$ is a legitimate type of sound speed in space.

In the IPM, $C_{s}$ ~ 13 - 240 km/s [e.g., Refs. 12; 33; 34].

Speed of magnetic fields

The cryptic title is alluding to what is known as the Alfvén speed, which is defined as: $$ V_{A} = frac{ B_{o} }{ sqrt{ mu_{o} rho } } $$ where $B_{o}$ is the magnitude of quasi-static, ambient magnetic field, $mu_{o}$ is the permeability of free space, and $rho$ is the plasma mass density (which is roughly equivalent to the ion mass density unless it's a pair plasma). This speed is typically associated with transverse Alfvén waves, but the speed is relevant to information transfer in plasmas, thus why I included it here.

In the IPM, $V_{A}$ ~ 4 - 220 km/s [e.g., Refs. 10; 12; 33; 34].

Speed of magnetized sound waves

In a magnetized fluid like a plasma, there are fluctuations that are compressive whereby they compress the magnetic field in phase with the density. These are known as magnetosonic or fast mode waves. The full MHD definition of the phase speed for a fast mode wave is given by: $$ 2 V_{f}^{2} = left( C_{s}^{2} + V_{A}^{2} right) + sqrt{ left( C_{s}^{2} + V_{A}^{2} right)^{2} + 4 C_{s}^{2} V_{A}^{2} sin^{2}{theta} } $$ where $theta$ is the angle of propagation with respect to $mathbf{B}_{o}$. $V_{f}$ is the relevant speed for shock waves in weakly collisional and collisionless plasmas. It is also a type of sound speed, thus the name magnetosonic.

In the IPM, $V_{f}$ ~ 17 - 300 km/s [e.g., Refs. 10; 12; 33; 34].

Side Note
There is also a slow mode wave, which differs in polarization and the relative phase between the magnetic and density fluctuations. It is called slow because it has a smaller phase speed than the fast mode in the same medium.

Thermal speeds

The last two speeds that are relevant are the thermal speeds of the electrons and ions. The one-dimensional rms speed is given by: $$ V_{Ts}^{rms} = sqrt{frac{ k_{B} T_{s} }{ m_{s} }} $$ where the definitions are the same as in previous sections and $s$ can be $e$(electrons) or $i$(ions). Generally we use the three dimensional most probable speed, which is given by: $$ V_{Ts}^{mps} = sqrt{frac{ 2 k_{B} T_{s} }{ m_{s} }} $$

In the IPM, the electron [e.g., Refs. 2; 3; 5; 7; 8; 14; 17-22; 24; 25; 27; 29-34] and ion [e.g., Refs. 1-6; 8-11; 13; 15-17; 19; 20; 23; 26-32] thermal speeds are $V_{Te}^{mps}$ ~ 1020 - 5170 km/s and $V_{Ti}^{mps}$ ~ 13 - 155 km/s, respectively.

Summary

There are several different types of sound-like speeds in space and each of them can produce similarly related phenomena. For instance, we often refer to Mach numbers associated with $C_{s}$, $V_{A}$, and $V_{f}$. In addition, there are several plasma instabilities that result from an effect that similar to Cerenkov radiation, whereby a beam of particle exceeds, for instance, the electron thermal speed.

In summary, in the regions outside of local magnetospheres but within the realm of our sun's influence, there is a wide range of sound speeds.

Update

A paper on the statistics of temperature-dependent parameters near Earth in the solar wind was recently published in Astrophys. J. Suppl. by Wilson et al. [2018] (it's Open Access so no paywall). The work provides new measurements but also provides a detailed literature review/reference list of past work.

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Direct measurement shows the speed is 1100 m/s.

ESA's dart-like Gravity field and Ocean Circulation Explorer (GOCE) Earth Explorer used to orbit as close to Earth as possible - just 260 km up - to maximize its sensitivity to variations in Earth's gravity field. At that altitude, there is enough atmosphere to exert a small drag. The satellite had an aerodynamic shape and a small engine to keep it in orbit. The mission ended when the engine ran out of fuel.

In 2011, the huge 9.1 Japanese Tohoku earthquake generated atmospheric disturbances. These deflected the satellite. Density variations were also measured. Article and video here.