A rocket that can launch a single cubesat would have to be competitive in the current market. But that market is skewed: cubesats are passengers on launches that usually have one large satellite (the main payload) plus a bunch of cubesats/smallsats. So the launch cost is shared. This consists of not only the cost to build the rocket, but there's development cost and launch services (as mentioned by GremlinRanger). Because a cubesat is only 1/1000 the weight of the main payload, the launch price is only 1/1000 of the main payload (on the order of $100,000). This sets a ceiling for the launch price you can set for your rocket.
So your launch can't cost more than $100k, and this has to pay for everything: building the rocket, development cost and launch services.
Is that a feasible target? Let's look at some small rockets that are produced in large numbers: military missiles. An AIM-120 AMRAAM air-air missile with a top speed of ~Mach 4 (or 1/5 of the speed you need, 1/25 of the energy you need) costs \$300k and is produced by the thousands. Now part of that is for parts you don't need (guidance, warhead, ability to be launched from a fighter aircraft). On the other hand, as a rocket company you have less history to build on. The military has trouble developing missiles that cost less than \$100k each.
Small rockets are cheaper than large ones, but some parts of the development are still expensive: you still need a high-performance rocket engine (probably 2, to get a 2-stage rocket). If 10% of your launch price goes toward paying off your development cost, you have to launch thousands of rockets to pay for the development cost.
At least one attempt has been made to develop a rocket in the one-cubesat class: the Japanese SS-520. Development cost was $3.5 million, and it was based on an existing sounding rocket. While test launches were successful, there don't seem to be plans to put this into production.
Earlier in this answer I mentioned a cubesat launch can't cost more than \$100k. The Japanese estimated they could set the price at $500k because a dedicated launch offers added value: the sat can be launched into a specified orbit instead of being tied to the orbit of a main payload. That changes the financial picture. At the same time though, companies that sell piggyback launches can develop upper stages that give similar options to launch into specified orbits.