When you say an efficient portfolio, I assume you mean a portfolio that lies on the efficient frontier, which is the most efficient portfolio in terms of risk-return trade-off, when we have only risky assets to chose from.
Different points/portfolios on the efficient frontier have different levels of risk and therefore different beta values. However, the 'market' is just one of these efficient portfolios and is determined by the point at which a line starting from the risk-free rate of return on the y-axis is tangential to the efficient frontier, and this then determines the risk-level that is equivalent to a beta of unity. The other efficient portfolios can then be assessed in relation to this one market portfolio in terms of the percentage of that market risk level.
However, once the market portfolio has been determined in this way, no rational person would invest in another portfolio on the efficient frontier. This is because we now have a set of portfolios that supersedes the efficient frontier for all levels of risk except that of the market portfolio (where the line through the risk-free rate touches the efficient frontier). This more efficient set of portfolios is the infinite set of combinations between just two assets - the risk-free asset and the one market portfolio asset. If you put some of your money on deposit at the risk-free rate and the rest of your money in the one market portfolio then you will obtain a risk-return trade-off that is somewhere along the line between the risk-free rate and the market portfolio, i.e. your beta will be somewhere in between zero and 1. The minute you borrow money at the risk-free rate and invest your own money plus the borrowed money in the one market portfolio then you will obtain a risk-return trade-off that is somewhere along the same* line but beyond the point representing the one market portfolio, this is what makes your beta greater than one - you are gearing yourself to the market. The line joining the risk-free rate with the one market portfolio is what is called the 'Capital Market Line'.
- Actually, it is not quite the same line, because the risk-free borrowing rate is in practice higher than the risk-free lending rate and so as compared to the line drawn from a mid-rate risk-free rate, the portion of the line to the left hand side of the market portfolio has a slightly lower starting point (and a slightly steeper gradient) and the portion beyond the market portfolio drops down a little (having a less steep gradient). The Capital Market Line is in practice a kinked line.
Hope that helps.