When there is an increase in the demand for money while the supply of money remains the same, there will typically be an increase in the rate of interest. This is because individuals and businesses are willing to pay a higher interest rate to borrow money when there is increased demand for it. As a result, the cost of borrowing money increases, leading to a higher rate of interest.

It is important to note that the other options mentioned in the question (fall in the level of prices, decrease in the rate of interest, increase in the level of income and employment) are not necessarily direct consequences of an increase in the demand for money while the money supply remains constant. The impact on prices, interest rates, and the economy as a whole will depend on various other factors and economic conditions.